He creado unToolbar para Lollipop, pero parece que no puedo entender cómo agregar un botón de desbordamiento a la barra de herramientas. No voy a usar la barra de herramientas v7 appcompat, ya que quiero usar explícitamenteToolbar widget Quiero que se vea así:

main_activity.xml

<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:fab="http://schemas.android.com/apk/res-auto"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:orientation="vertical">

<Toolbar xmlns:android="http://schemas.android.com/apk/res/android"
    android:id="@+id/toolbar"
    android:layout_width="match_parent"
    android:layout_height="112dp"
    android:elevation="4dp"
    android:layout_alignParentTop="true"
    android:minHeight="?android:attr/actionBarSize"
    android:background="?android:attr/colorPrimary"
    android:gravity="bottom"
    android:paddingBottom="16dp"/>
</RelativeLayout>

Clase principal

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_quotebook);


    getWindow().addFlags(WindowManager.LayoutParams.FLAG_DRAWS_SYSTEM_BAR_BACKGROUNDS);
    getWindow().setStatusBarColor(getResources().getColor(R.color.primary_dark));
    getWindow().setNavigationBarColor(getResources().getColor(R.color.primary_dark));

    initToolbar();

}

private void initToolbar() {
    mToolbar = (Toolbar) findViewById(R.id.toolbar);
    mToolbar.setTitleTextColor(Color.WHITE);
    mToolbar.setTitle(R.string.app_shortname);
    mToolbar.showOverflowMenu();
}

@Override
public boolean onCreateOptionsMenu(Menu menu) {
    getMenuInflater().inflate(R.menu.menu_overflow, menu);
    return true;
}

@Override
public boolean onOptionsItemSelected(MenuItem item) {
    // Handle action bar item clicks here. The action bar will
    // automatically handle clicks on the Home/Up button, so long
    // as you specify a parent activity in AndroidManifest.xml.
    int id = item.getItemId();

    //noinspection SimplifiableIfStatement
    if (id == R.id.action_settings) {
        return true;
    }

    return super.onOptionsItemSelected(item);
}

menu.xml

<menu xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto">

<item
    android:id="@+id/action_settings"
    android:orderInCategory="100"
    android:title="@string/action_settings"
    android:showAsAction="always"
    android:visible="true"/>

</menu>