1 tablespoon vegetable oil, plus more for the pizzas and grill grates
1 pound bulk breakfast sausage
2 tablespoons all-purpose flour, plus more for dusting
2 1/2 cups whole milk
Kosher salt and freshly ground pepper
1 pound frozen pizza dough, thawed
8 ounces sharp white Cheddar, shredded (about 3 heaping cups)
4 large eggs
2 scallions, thinly sliced
Hot sauce, for serving