Thirty minutes before you plan to make the pizza, position a rack at the lowest level in the oven and put a pizza stone on it. Preheat the oven to 500 degrees F.
Cook the bacon in a large skillet over medium heat, turning as needed, until almost crisp, about 10 minutes. Transfer the bacon to a paper towel-lined plate. Let cool, then crumble and set aside. Reserving the bacon drippings in the skillet.
Meanwhile, place the broccoli in a mesh strainer or fine colander and run cold water on it to thaw. Set it over a bowl to strain out excess water. Put it in a medium bowl, pour 2 tablespoons of bacon drippings over it, sprinkle with some salt and pepper and toss to coat; set aside.
Dust the work surface with a little flour. Put the parchment paper on a pizza peel or the back of a baking sheet. Halve the pizza dough and cover one half with a towel. Drape the other half over your fist and gently tug and turn it, letting the weight of the edges help stretch it thinner, until you have a round about 1/4-inch thick and 12 inches across. Place the dough on the parchment and scatter with 1 cup of the Cheddar, then half the broccoli, leaving a 1-inch border. Slide the parchment onto the hot pizza stone and bake for 5 minutes.
Crack 1 egg into each of three small bowls. After the pizza has baked for 5 minutes, rotate it and slide the eggs onto it, spacing evenly. Scatter 1/2 cup of the remaining cheese and half the bacon on top. Bake until the pizza is browned and crisp on the bottom, the egg whites are set and the yolks are still runny, about 5 minutes. Transfer the pizza to a cutting board and let sit for 2 minutes to allow the eggs to set. Repeat with the other half of the dough and the remaining ingredients. Slice and serve with crushed red pepper flakes or hot sauce.