Peel and halve the onions. Slice them evenly into thin half moons, about 1/8 inch thick.

Melt butter with olive oil in a wide heavy pot over medium heat. Add onions and salt. Cook, stirring often, until softened, then reduce heat to medium-low. Continue cooking 45–60 minutes in total, stirring more frequently as they brown, until soft and deeply golden. If the pan darkens too quickly, add a small splash of water and scrape gently; do not let the onions burn.

Add the 1/2 cup water and scrape the flavorful browned bits from the bottom. Pour in broth; add thyme, bay leaf and pepper. Bring to a gentle simmer and cook uncovered for 20 minutes. Remove the bay leaf and taste before adding more salt.

While soup simmers, heat the oven to 400°F (200°C). Arrange baguette slices on a baking sheet and toast for 6–10 minutes, turning once, until dry and lightly golden.

Switch the oven to broil. Divide hot soup among four broiler-safe crocks on a baking sheet. Add two toast slices to each and divide the grated cheese over them. Broil until cheese bubbles and browns, about 2–4 minutes, watching constantly. If crocks are not broiler-safe, broil cheese on the toast separately and add to bowls.

Use oven mitts and place hot crocks on heatproof plates. Let cool briefly before eating. Add a few thyme leaves if desired.