Fix WidgetsBinding.firstFrameRasterized not completing (#56022)

packages/flutter/lib/src/widgets/binding.dart

@@ -877,6 +877,9 @@ mixin WidgetsBinding on BindingBase, ServicesBinding, SchedulerBinding, GestureB
     }
     _needToReportFirstFrame = false;
     if (firstFrameCallback != null && !sendFramesToEngine) {
+      // This frame is deferred and not the first frame sent to the engine that
+      // should be reported.
+      _needToReportFirstFrame = true;
       SchedulerBinding.instance.removeTimingsCallback(firstFrameCallback);
     }
   }

packages/flutter/test/widgets/binding_first_frame_rasterized_test.dart

+// Copyright 2014 The Flutter Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+import 'dart:ui';
+
+import 'package:flutter/material.dart';
+import 'package:flutter_test/flutter_test.dart';
+import 'package:quiver/testing/async.dart';
+
+void main() {
+  test('Deferred frames will trigger the first frame callback', () {
+    FakeAsync().run((FakeAsync fakeAsync) {
+      final WidgetsBinding binding = WidgetsFlutterBinding.ensureInitialized();
+      binding.deferFirstFrame();
+
+      runApp(const Placeholder());
+      fakeAsync.flushTimers();
+
+      // Simulates the engine completing a frame render to trigger the
+      // appropriate callback setting [WidgetBinding.firstFrameRasterized].
+      binding.window.onReportTimings(<FrameTiming>[]);
+      expect(binding.firstFrameRasterized, isFalse);
+
+      binding.allowFirstFrame();
+      fakeAsync.flushTimers();
+
+      // Simulates the engine again.
+      binding.window.onReportTimings(<FrameTiming>[]);
+      expect(binding.firstFrameRasterized, isTrue);
+    });
+  });
+}