Fix menu anchor state handling (#157612)

This commit refactors the `_MenuAnchorState` class in `menu_anchor.dart` to include a check for the mounted state and the scheduler phase before calling `setState()`. This ensures that UI updates are only performed when the widget is still mounted and not during the persistent callbacks phase. Additionally, a new test case is added in `menu_anchor_test.dart` to verify that the `isOpen` state of the `MenuAnchor` widget is updated correctly when the button is pressed.

Fix: #157606

*Replace this paragraph with a description of what this PR is changing or adding, and why. Consider including before/after screenshots.*

*List which issues are fixed by this PR. You must list at least one issue. An issue is not required if the PR fixes something trivial like a typo.*

*If you had to change anything in the [flutter/tests] repo, include a link to the migration guide as per the [breaking change policy].*

packages/flutter/lib/src/material/menu_anchor.dart

@@ -282,6 +282,9 @@ class MenuAnchor extends StatefulWidget {
   ///
   /// If not supplied, then the [MenuAnchor] will be the size that its parent
   /// allocates for it.
+  ///
+  /// If provided, the builder will be called each time the menu is opened or
+  /// closed.
   final MenuAnchorChildBuilder? builder;
 
   /// The optional child to be passed to the [builder].
@@ -576,6 +579,11 @@ class _MenuAnchorState extends State<MenuAnchor> {
     }
 
     widget.onOpen?.call();
+    if (mounted && SchedulerBinding.instance.schedulerPhase != SchedulerPhase.persistentCallbacks) {
+      setState(() {
+        // Mark dirty to ensure UI updates
+      });
+    }
   }
 
   /// Close the menu.

packages/flutter/test/material/menu_anchor_test.dart

@@ -4737,6 +4737,63 @@ void main() {
     final MenuController controller = MenuController();
     expect(controller.isOpen, false);
   });
+
+  // Regression test for https://github.com/flutter/flutter/issues/157606.
+  testWidgets('MenuAnchor updates isOpen state correctly', (WidgetTester tester) async {
+    bool isOpen = false;
+    int openCount = 0;
+    int closeCount = 0;
+
+    await tester.pumpWidget(
+      MaterialApp(
+        home: Scaffold(
+          body: Center(
+            child: MenuAnchor(
+              menuChildren: const <Widget>[
+                MenuItemButton(child: Text('menu item')),
+              ],
+              builder: (BuildContext context, MenuController controller, Widget? child) {
+                isOpen = controller.isOpen;
+                return FilledButton(
+                  onPressed: () {
+                    if (controller.isOpen) {
+                      controller.close();
+                    } else {
+                      controller.open();
+                    }
+                  },
+                  child: Text(isOpen ? 'close' : 'open'),
+                );
+              },
+              onOpen: () => openCount++,
+              onClose: () => closeCount++,
+            ),
+          ),
+        ),
+      )
+    );
+
+    expect(find.text('open'), findsOneWidget);
+    expect(isOpen, false);
+    expect(openCount, 0);
+    expect(closeCount, 0);
+
+    await tester.tap(find.byType(FilledButton));
+    await tester.pump();
+
+    expect(find.text('close'), findsOneWidget);
+    expect(isOpen, true);
+    expect(openCount, 1);
+    expect(closeCount, 0);
+
+    await tester.tap(find.byType(FilledButton));
+    await tester.pump();
+
+    expect(find.text('open'), findsOneWidget);
+    expect(isOpen, false);
+    expect(openCount, 1);
+    expect(closeCount, 1);
+  });
 }
 
 List<Widget> createTestMenus({