Caramelize the Onions:

1. In a large, heavy-bottomed pot, heat the olive oil and butter over medium heat until the butter is melted and begins to bubble.
2. Add the sliced white and red onions to the pot, stirring to coat them in the oil and butter.
3. Reduce the heat to medium-low and cook the onions slowly, stirring occasionally, for about 25-30 minutes or until they become golden brown and caramelized.
4. Take care not to burn the onions, adjusting the heat as necessary.

Thicken with Flour:

1. Once the onions are caramelized and soft, sprinkle the flour over them and stir to combine.
2. Cook the flour for about 2-3 minutes to remove its raw taste, stirring constantly to prevent it from sticking to the bottom of the pot.

Add the Liquid:

1. Slowly pour in the white wine, stirring to deglaze the pan and scrape up any bits stuck to the bottom.
2. Let the wine simmer for about 5 minutes to cook off some of the alcohol.
3. Add the water and beef bouillon, stirring to dissolve the bouillon.
4. Bring the soup to a simmer and let it cook for 15-20 minutes, allowing the flavors to meld together.

Prepare the Baguette:

1. While the soup simmers, preheat your oven’s broiler.
2. Lay the baguette slices on a baking sheet and toast them under the broiler for about 2-3 minutes on each side, until golden brown.

Assemble and Broil the Soup:

1. Preheat your oven to broil.
2. Ladle the hot soup into oven-safe bowls.
3. Place one or two slices of toasted baguette on top of each bowl of soup, ensuring it covers most of the surface.
4. Generously sprinkle the shredded Gruyère and grated Parmesan cheeses over the baguette slices.
5. Place the bowls under the broiler for 3-4 minutes or until the cheese is melted, bubbly, and golden brown.

Serve:

1. Carefully remove the bowls from the oven (they will be hot!).
2. Serve the soup immediately, garnished with a bit more cheese if desired.
3. This French onion soup pairs beautifully with a light salad or crusty bread on the side.