Heat butter in a medium nonstick skillet over medium heat. Add diced red onion and pepper, sauté until softened, about 2 minutes.
Add the eggs and cook, stirring, until just set, about 2-3 minutes. Remove from heat. Season with salt and pepper.
Warm tortillas: Heat whole wheat tortillas in a skillet over medium heat for 15-20 seconds per side, or microwave them wrapped in a damp paper towel for 20 seconds, until pliable.
Assemble burritos: Lay each tortilla flat. Portion the eggs and black beans down the center. Top with cheese optional. Fold in the sides and roll up. Cut in half.