Combine one-third of onion slices and 2 tablespoons butter in a large skillet; sauté until onion is transparent. Transfer onions to a 5-quart casserole. Repeat procedure with remaining onion slices and butter.
Add 2 cups broth to onions in casserole, and bring to a boil. Reduce heat; cover and simmer 20 minutes or until onion is tender. Add remaining broth, and bring to a boil; remove soup from heat.
Cut each slice of French bread in half; toast each side. Arrange toasted bread on top of soup in casserole; sprinkle with cheese. Broil 8 inches from heating element 3 minutes or until cheese melts and becomes golden brown.
Ladle into individual soup bowls, placing a piece of toast in each bowl. Serve immediately.