Robust solution (can be used as a snippet): Take an event and then get control that fired that event. Use that control to get the Stage:
@FXML
private void browseDirectory(ActionEvent event) {
Stage stage = Stage.class.cast(Control.class.cast(event.getSource()).getScene().getWindow());
DirectoryChooser directoryChooser = new DirectoryChooser();
File selectedDirectory = directoryChooser.showDialog(stage);
System.out.println(selectedDirectory.getAbsolutePath());
}