Melt the butter in a large nonstick skillet over medium-low heat. Pour in the egg mixture and let sit until the eggs just begin to place around the edges, about 2 minutes. With a rubber spatula, push the set eggs in the edges into the center. Distribute the uncooked eggs into an even coating. Duplicate, pushing the place eggs from the edges into the center every 30 seconds until almost set, for a total cooking time of 6 to 8 minutes. (The top of the eggs should still be slightly moist.) Transfer the eggs to a bowl.
To assemble the sandwiches, spread about 1 tsp of the mustard over the cut-side of every croissant underside. Place a slice of ham over the bottom piece of each croissant. Top each with a heaping 1/2 cup of scrambled eggs along with 1/4 cup of cheese. Set the top half of each croissant within the sandwich.
To serve immediately: Arrange a rack in the center of the oven and heat to 350°F. Place the sandwiches on a baking sheet. Bake until the cheese is melted, 8 to 10 minutes.
Reheating: If not serving immediately, tightly wrap each sandwich with aluminum foil and keep in resealable bags in the fridge or freezer. Reheat uncovered in a 325°F oven until warmed through, about 20 minutes if refrigerated, roughly 30 minutes if frozen.
Storage: The foil-wrapped sandwiches could be kept in a resealable plastic bag in the refrigerator for up to 3 days or in the freezer for up to 3 weeks.