Hamburger Steak with Creamy Onion Gravy

1 large Vidalia or sweet onion, halved and sliced (about 3 cups)
1 cup beef broth
1 cup water
¼ cup all-purpose flour
¼ teaspoon seasoned salt, or to taste
¼ teaspoon freshly ground black pepper, or to taste
½ tablespoon Worcestershire sauce (optional)
1 teaspoon browning & seasoning sauce (like Kitchen Bouquet, optional)
2 cups sliced mushrooms (optional)
Instructions: