Preheat your oven to 375°F (190°C).

In a large skillet, heat the olive oil over medium heat. Add the sliced onions and stir occasionally until they become caramelized, about 30 minutes. Add the minced garlic during the last few minutes.

In a separate bowl, combine the vegetable broth and heavy cream. Stir in the thyme, salt, and pepper.

In a large baking dish, layer half of the cubed bread at the bottom. Pour half of the onion mixture over the bread, then sprinkle with half of the Swiss and mozzarella cheese. Repeat these layers, finishing with the remaining cheese on top.

Carefully pour the vegetable broth mixture over the entire casserole, ensuring that all the bread is soaked.

Cover the dish with aluminum foil and bake in the preheated oven for about 30 minutes. Remove the foil and bake for an additional 15-20 minutes, or until the cheese is bubbly and golden brown.

Remove from the oven and let it cool for a few minutes. Garnish with fresh parsley, if desired, and serve warm.