Caramelize the onions until rich brown.
Cut or tear the sourdough bread into bite-sized chunks and toss with olive oil, then air-fry or bake until crisp.
Grate the Gruyère and Parmesan cheese.
Add butter to the caramelized onions and stir in a spoonful of flour, letting it brown lightly.
Add stock and bring to a boil, adjusting the amount based on desired soupiness.
Lower heat to a simmer and add cooked beans.
Season with salt, pepper, Worcestershire sauce, and vinegar.
Spoon toasted bread onto the beans and top with grated cheese.
Cover the pot to melt the cheese or broil for a browned finish.
Serve with a crack of pepper and a sprinkle of chives if desired.