Preheat oven to 350 degrees F. Grease an 8x8 baking dish and set aside.
Mix the Swiss and Gruyere cheese together.
In a medium skillet over medium heat, heat 2 Tbsp butter. When the pan is hot and butter has melted, add the eggs, and stir. Remove from heat while eggs are still glossy and wet—you want the eggs to be slightly undercooked. Cover and set aside.
Lay out your tortillas on a flat surface. Place 2 slices of ham on each. Spread about 2 1/2 tsp of jam on top, followed by 1/4 of the eggs and 1/4 of cheese. Season with fresh pepper, then roll each burrito and spread 1 tsp mayonnaise onto the bottom of each. Place in the pan, seam side down, and sprinkle with remaining cheese and fresh cracked pepper. Place a piece of parchment over the top, then cover with foil and bake for 25-30 minutes until the cheese has melted, and the burrito bottoms have lightly browned. Continue baking for 8-10 minutes until the cheese has browned and begins to crisp on the bottom of the pan. If the tortillas start to become too crispy, turn on the broiler to fully melt the cheese on top.
Remove from the oven and serve hot with a sprinkle of powdered sugar.