Place a pizza stone or inverted rimmed baking sheet on the lowest rack of the oven and preheat to 400 degrees F.
Set a cooling rack inside another rimmed baking sheet, spread out the bacon and bake on the stone or baking sheet until cooked and crispy, 20 to 22 minutes. Crumble and set aside on a paper towel-lined plate.
Increase the oven temperature to 500 degrees F.
Dust a piece of parchment lightly with flour. Stretch or roll out the pizza dough into a 12-inch round on the parchment, then transfer the dough on the parchment to a pizza peel or another inverted rimmed baking sheet. Sprinkle with the Cheddar and bacon. Brush the crust with the butter and sprinkle with the bagel seasoning.
Slide the pizza on the parchment paper onto the stone or baking sheet and bake until the cheese is melted and the crust is almost golden, about 6 minutes. Remove from the oven.
Using the back of a spoon, make 5 small wells in the cheese and bacon and break an egg into each of the wells. Bake until the eggs are just set, about 5 minutes.
Garnish the pizza with the chives and serve with hot sauce.
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