Preheat oven to 475 degrees F (245 degrees C).

Cook and stir sliced Italian sausage in a skillet over medium heat until some of the fat has rendered out and sausage is no longer pink inside, about 5 minutes. Transfer sausage slices to a paper towel-lined plate to drain.

Roll pizza dough out to about 12 inches in diameter and about 3/8-inch thick; sprinkle with cornmeal. Flip dough so the cornmeal side is down and transfer to a perforated 14-inch pizza pan.

Spread pizza sauce in a thin layer on top of the dough. Sprinkle with red pepper flakes and about 3/4 the Fontina cheese. Spread sausage slices onto the pizza, leaving 4 spaces about 3 inches across on the pizza for eggs. Sprinkle remaining Fontina cheese onto sausage slices.

Bake in the preheated oven until dough is hot, edges are starting to brown, and cheese has started to melt, 10 to 12 minutes.

Crack each egg into a separate ramekin or small bowl. Remove pizza from oven and use a spatula to press the spaces between sausage slices flat. Gently pour an egg onto each space. Grind a little black pepper onto each egg yolk. Lightly sprinkle Parmigiano-Reggiano cheese over entire pizza.

Return to oven and bake until eggs are set but yolks are still liquid, about 5 minutes, checking frequently to avoid overcooking eggs. Toss arugula leaves with olive oil in a bowl until coated. Remove pizza from oven, poke egg yolks with a fork, spread them around a little, and sprinkle arugula over the top of the pizza. Cut into 4 quarters, each with an egg, to serve.