In a large bowl, whisk together the eggs, parsley and hot sauce until combined. Season with a pinch of salt and pepper. Set aside.
Melt the butter in a large ovenproof nonstick skillet over medium-high heat. When the butter has melted, add the cubed ham and sweet potato, then cook for a couple minutes. Pour over the egg mixture and allow to cook for 30 to 45 seconds to set the edges. Place in the oven and bake until the eggs are set but not brown on top, 10 to 12 minutes. Set aside to cool slightly, 5 to 10 minutes, then slide the frittata out onto a cutting board and cut into 9 somewhat equal pieces.
Spread the mustard on the bottom half of each muffin and the mayo on the top half. To each bottom, add a piece of the frittata, a slice of each cheese and 1/3 cup of arugula. Top each with a muffin top. Serve with more hot sauce if desired.
To save time and stress in the morning, you can bake the frittata the night before. Allow the frittata to cool completely before wrapping and refrigerating overnight. To reheat, slice and pop in the microwave for 30 to 45 seconds.