好的，首先您必须稍微更改代码。

从更改Stateless为Stateful来管理小部件的状态，并放置一个名为的全局变量isLoading。您可以使用该变量，isLoading在按下按钮时将其设置为true，在按钮isLoading完成后将其设置为false。

在您的build方法内添加一个验证，以在isLoading为true 时显示循环进度，否则显示您的字段。

代码示例在这里：

    class ForgotPasswordScreen extends StatefulWidget {
      @override
      ForgotPasswordScreenState createState() {
        return new ForgotPasswordScreenState();
      }
    }

    class ForgotPasswordScreenState extends State<ForgotPasswordScreen> {
      final emailController = new TextEditingController();
      final authHandler = new Auth();
      bool isLoading = false;

      @override
      Widget build(BuildContext context) {
        return new Scaffold(
            body: Container(
                height: MediaQuery.of(context).size.height,
                decoration: BoxDecoration(
                  color: Colors.white,
                ),
                child: isLoading
                    ? Center(
                        child: CircularProgressIndicator(),
                      )
                    : new Column(
                        crossAxisAlignment: CrossAxisAlignment.center,
                        mainAxisSize: MainAxisSize.max,
                        mainAxisAlignment: MainAxisAlignment.center,
                        children: <Widget>[
                          new Row(
                            children: <Widget>[
                              new Expanded(
                                child: new Padding(
                                  padding: const EdgeInsets.only(left: 40.0),
                                  child: new Text(
                                    "EMAIL",
                                    style: TextStyle(
                                      fontWeight: FontWeight.bold,
                                      color: Colors.redAccent,
                                      fontSize: 15.0,
                                    ),
                                  ),
                                ),
                              ),
                            ],
                          ),
                          new Container(
                            width: MediaQuery.of(context).size.width,
                            margin: const EdgeInsets.only(
                                left: 40.0, right: 40.0, top: 10.0),
                            alignment: Alignment.center,
                            decoration: BoxDecoration(
                              border: Border(
                                bottom: BorderSide(
                                    color: Colors.redAccent,
                                    width: 0.5,
                                    style: BorderStyle.solid),
                              ),
                            ),
                            padding: const EdgeInsets.only(left: 0.0, right: 10.0),
                            child: new Row(
                              crossAxisAlignment: CrossAxisAlignment.center,
                              mainAxisAlignment: MainAxisAlignment.start,
                              children: <Widget>[
                                new Expanded(
                                  child: TextField(
                                    controller: emailController,
                                    textAlign: TextAlign.left,
                                    decoration: InputDecoration(
                                      border: InputBorder.none,
                                      hintText: 'PLEASE ENTER YOUR EMAIL',
                                      hintStyle: TextStyle(color: Colors.grey),
                                    ),
                                  ),
                                ),
                              ],
                            ),
                          ),
                          Divider(
                            height: 24.0,
                          ),
                          new Container(
                            width: MediaQuery.of(context).size.width,
                            margin: const EdgeInsets.only(
                                left: 30.0, right: 30.0, top: 20.0),
                            alignment: Alignment.center,
                            child: new Row(
                              children: <Widget>[
                                new Expanded(
                                  child: new FlatButton(
                                    shape: new RoundedRectangleBorder(
                                      borderRadius: new BorderRadius.circular(30.0),
                                    ),
                                    color: Colors.redAccent,
                                    onPressed: () {
                                      setState(() {
                                        isLoading = true;
                                      });
                                      authHandler
                                          .sendPasswordResetEmail(
                                              emailController.text)
                                          .then((void nothing) {
                                        print("done");
                                        setState(() {
                                          isLoading = false;
                                        });
                                      }).catchError((e) => print(e));
                                    },
                                    child: new Container(
                                      padding: const EdgeInsets.symmetric(
                                        vertical: 20.0,
                                        horizontal: 20.0,
                                      ),
                                      child: new Row(
                                        mainAxisAlignment: MainAxisAlignment.center,
                                        children: <Widget>[
                                          new Expanded(
                                            child: Text(
                                              "FORGOT PASSWORD",
                                              textAlign: TextAlign.center,
                                              style: TextStyle(
                                                  color: Colors.white,
                                                  fontWeight: FontWeight.bold),
                                            ),
                                          ),
                                        ],
                                      ),
                                    ),
                                  ),
                                ),
                              ],
                            ),
                          ),
                        ],
                      )));
      }
    }