Preheat oven to 200ºF. Arrange flatbreads in a single layer on a baking sheet and place in oven to warm.

In a medium skillet, heat 2 tablespoons butter until foaming. Add peppers, onions, and ham and cook until vegetables have softened, about 3 minutes. Transfer to a bowl and wipe out skillet.

Meanwhile, whisk eggs, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a large bowl. Melt remaining butter in skillet over medium heat until foaming. Add eggs and gently stir until the eggs have set, about 2 to 3 minutes.

To assemble the wraps, remove warmed flatbreads from oven. Divide the scrambled eggs and ham mixture among all flatbreads. Top each wrap with 2 tablespoons shredded cheese. Roll and slice in half to serve.