Sprinkle steak with 1/2 teaspoon salt and 1/2 teaspoon pepper.

Heat butter and oil in a 6-quart Dutch oven or heavy-bottomed pot over medium-high heat. Add steak; sear for 2 to 3 minutes per side. Remove steak; drain all but 1 tablespoon drippings. Stir in onions and thyme; cook, stirring occasionally, until onions are tender, about 15 minutes.

Add wine and bring to a boil, stirring with a wooden spoon to scrape up any browned bits. Boil until wine is reduced by half, about 15 minutes.

Cut steak from the bones and chop into 1-inch pieces; transfer to the pot. Add broth and bring to a boil. Reduce heat; simmer, uncovered, stirring occasionally, until beef is very tender, about 3 hours. Add water as needed to keep onions and beef covered.

Preheat the oven to 425 degrees F (220 degrees C). Arrange bread on a small baking sheet.

Bake bread in the preheated oven until toasted, about 6 minutes.

Stir mozzarella cheese and Parmesan cheese together in a small bowl.

Remove and discard bones and thyme sprigs from the pot. Skim off any excess fat. Stir in remaining salt and pepper.

Set an oven rack 4 to 6 inches from the heat source and preheat the oven's broiler. Divide soup among 4 oven-proof bowls or mugs; arrange on a large foil-lined baking sheet. Top each bowl with toast and cheeses.