Adjust oven rack to middle position and preheat oven to 400°F (205°C).

In a large skillet, melt butter over medium-high heat until foaming.

Add onions, salt, and pepper, and cook, stirring occasionally, until softened, 6 to 8 minutes.

Reduce heat to medium-low and cook, stirring frequently and scraping bottom of skillet with wooden spoon as needed to avoid scorching, until onions are sweet and deeply caramelized, about 45 minutes.

If browned onion juices on bottom of skillet threaten to burn, add 1 tablespoon (15 ml) water, scrape up browned bits, and continue cooking.

While onions caramelize, arrange bread cubes in an even layer on a rimmed baking sheet and bake, stirring halfway through, until lightly toasted, 8 to 10 minutes.

Remove from oven and set aside. Set broiler to high.

Once onions are caramelized, add garlic and thyme to the onions and cook, stirring constantly, until fragrant, about 1 minute.

Add sherry, scraping up any browned bits with a wooden spoon, and cook over medium heat until alcohol is mostly evaporated, 1 to 2 minutes.

Add beans and broth and bring to a simmer over medium-high heat.

Cook, stirring often, until beans are tender and broth is slightly thickened, about 7 minutes.

Season to taste with salt and pepper. Remove from heat.

Top bean mixture evenly with toasted bread cubes, then sprinkle Gruyère and Parmigiano-Reggiano evenly over top.

Broil until cheese is melted and browned in spots, about 2 minutes.

Garnish with thyme and black pepper. Serve immediately.