1 tablespoon finely chopped fresh chives or scallions
Arrange a rack in the middle of the oven and heat to 400°F.
Heat 1 tablespoon of the butter in a 10-inch cast iron skillet over medium heat until bubbling. Add the bread cubes to the pan, toss to coat with the butter, and arrange in a single layer. Toast the bread, tossing every minute or so, until the bread cubes are golden-brown on all sides, about 5 minutes total. Transfer the croutons to a bowl; set aside.
Reduce the heat to low. Add the oil, onions, and 1/2 teaspoon of the salt to the same skillet. Cook, stirring every 5 to 10 minutes and scraping any browned build-up from the bottom of the pan, until the onions are soft and deeply browned, about 40 minutes total. Add the vinegar and scrape up the browned bits at the bottom of the pan.
Whisk together the eggs, milk, mustard, remaining 1 teaspoon salt, and pepper in a large bowl. Stir the remaining butter and croutons into the skillet, then spread in an even layer. Pour the egg mixture over the top. Tilt the pan to make sure the eggs settle evenly. Top with the cheese. Cook until the eggs at the edges of the skillet begin to set, 2 to 4 minutes.
Bake until the eggs are set, 8 to 10 minutes. To check, cut a small slit in the center of the frittata. If raw eggs run into the cut, bake for another few minutes; if the eggs are set, pull the frittata from the oven.
Cool in the pan for 5 minutes, top with chives, then slice into wedges and serve warm.