1. Preheat oven to 425°F (220°C).
Melt the butter in a cast iron skillet suitable for both stovetop cooking and the oven. Add the onions, 1/4 teaspoon salt and 1/8 teaspoon black pepper. Toss, using tongs, and cook over medium heat until the onions begin to relax. Toss again, and lower the temperature to medium-low. Cook for 20 minutes, or until caramelized. Remove to a bowl.
Measure 2 tablespoons olive oil into the pan, and add the Brussels sprouts, and the remaining salt and pepper. Toss to coat, and flip all the sprouts over so that their flat sides are in contact with the pan. They might not all fit, but do this with as many of the sprouts as possible. Cook over medium heat without moving them for 8 minutes. They should be well-seared.
Add the garlic paste, reserved caramelized onions, and balsamic vinegar. Toss well to coat, and add additional olive oil if the pan is completely dry. Cook on the stovetop over medium heat for a few minutes before sliding the skillet into the oven.
Roast for 15 minutes. If you plan to cook the eggs in the same pan, remove the pan and spread the roasted vegetables out to create two 3-inch circles on the bottom of the skillet. Pour just tiny bit of olive oil into each circle. Crack the eggs into a ramekins to ensure their yolks are not broken, and slide them into the circular openings in the roasted vegetables. Slide the skillet back into the oven, and bake for 7 minutes for runny eggs.
Remove and finish the dish with a little flourish of celery salt before serving.