Heat 1 tablespoon butter in a large oven-proof skillet over medium heat until melted. Add the bread and toss to coat. Spread in an even layer and toast the bread, tossing every minute or two until the bread cubes are golden brown on all sides, about 5 minutes. Remove bread from pan and set aside.

Reduce heat to low, add 1 tablespoon butter and 2 tablespoons olive oil. Heat until the butter is melted. Add the onions and ½ teaspoon of salt. Cook on low heat, stirring every 5 minutes until the onions are soft and brown, about 40 minutes. Add the vinegar and cook, stirring, until the vinegar has evaporated.

Preheat the oven to 400 degrees.

Whisk together the eggs, beef broth, mustard, Worcestershire sauce, 1 teaspoon salt, and pepper.

Add the remaining tablespoon of butter, let melt, then add the toasted bread cubes to the pan and toss to coat. Spread in an even layer. Pour the egg mixture over the onions and bread, tilting the pan to make sure the egg mixture spreads evenly. Sprinkle the cheese over the top and cook until the edges are set, 2-4 minutes. Transfer the pan to the oven and bake until the eggs are set, about 10 minutes.

Sprinkle with chives, cut into wedges and serve.