a 12- to 13-inch pizza stone or baking sheet; a pizza peel
Put a 12- to 13-inch pizza stone on the bottom rack in the oven and preheat to 500 degrees F for at least 30 minutes.
For the bechamel: Cook the diced bacon in a medium saucepan over medium heat until crispy, about 4 minutes. Transfer with a slotted spoon to a paper towel-lined plate to drain. Discard all but 2 tablespoons of the bacon drippings. Add the flour and whisk until combined. Continue to whisk until the mixture bubbles a bit, but does not brown, 1 to 2 minutes.
Slowly whisk in the milk until the mixture is smooth and there are no lumps. Reduce to low heat and cook, stirring occasionally, until the mixture coats the back of a spoon, 3 to 4 minutes. Season with salt and pepper.
For the pizza: Dust a work surface with the flour to help the pizza dough not stick. Press and roll out the pizza dough into a 10- to 12-inch round, 1/4-inch thick, leaving about 1 inch of the outside thicker for the crust.
Lightly dust a pizza peel with flour and put the dough on the peel. Spread the bechamel over the dough, leaving a 1/2-inch border. Sprinkle the grated fontina cheese on top of the bechamel. Carefully slip the dough onto the preheated stone and cook until the cheese is bubbly and golden, 8 to 10 minutes.
For the fried eggs: Meanwhile, fry the eggs. Melt 1 tablespoon butter in a large nonstick saute pan over high heat. Crack 3 eggs into the pan and cook until the bottom is crisp and the yolk sets, 4 minutes. Season with salt and pepper. Put the fried eggs on a sheet pan and repeat with the remaining 1 tablespoon butter and 3 eggs.
Remove the pizza from the oven and place in on a cutting board. Sprinkle the reserved chopped bacon and chopped chives over the pizza, then cut the pizza into 6 slices. Serve each slice topped with a crispy fried egg and a drizzle of the Calabrian chili oil, if desired.