## Background

The Windows runner has a race at startup:

1. **Platform thread**: creates a hidden window
2. **Platform thread**: launches the Flutter engine
3. **UI/Raster threads**: renders the first frame
4. **Platform thread**: Registers a callback to show the window once the next frame has been rendered.

Steps 3 and 4 happen in parallel and it is possible for step 3 to complete before step 4 starts. In this scenario, the next frame callback is never called and the window is never shown.

As a result the `windows_startup_test`'s test, which [verifies that the "show window" callback is called](https://github.com/flutter/flutter/blob/1f09a8662dad3bb1959b24e9124e05e2b9dbff1d/dev/integration_tests/windows_startup_test/windows/runner/flutter_window.cpp#L60-L64), can flake if the first frame is rendered before the show window callback has been registered.

## Solution

This change makes the runner schedule a frame after it registers the next frame callback. If step 3 hasn't completed yet, this no-ops as a frame is already scheduled. If step 3 has already completed, a new frame will be rendered, which will call the next frame callback and show the window.

Part of https://github.com/flutter/flutter/issues/119415

See this thread for alternatives that were considered: https://github.com/flutter/engine/pull/42061#issuecomment-1550080722