Arrange two racks in the oven—one in the top-middle position, the other in the bottom-middle. Preheat the oven to 400°F.
Remove the chilled dough rounds from the fridge. Peel off the top sheet of parchment from each one. Flip the dough over onto the counter or a baking sheet, then peel off the second sheet. Line two rimmed baking sheets with some of the parchment you just removed. Place two dough rounds on each sheet.
Spread each dough round with about 1/4 of the cheese mixture, leaving a 1-inch border around the edges. Top each with 1/4 of the kale–onion–corn mixture.
Fold the border of dough up and over the filling, pleating gently as you work your way around each galette.
Brush the dough border with egg wash. Sprinkle with Everything Bagel spice (or another topping of your choice).
Bake for 15 minutes. Then switch the trays—move the one from the bottom rack to the top, and vice versa. Bake for another 15 minutes.
Remove the pans from the oven and sprinkle each galette with the reserved fontina and Monterey Jack. Use a spoon to create a small well in the centre of each galette, then carefully crack an egg into each one. Return the pans to the oven and bake for 5–8 minutes more, until the whites are set but the yolks remain soft.
Just before serving, sprinkle each galette with salt, pepper, and a few dashes of hot sauce if desired.