In a medium bowl, whisk together the eggs and milk until well combined.
Heat a non-stick skillet over medium heat. Add the diced onion and bell pepper, cooking until softened, about 3-4 minutes.
Pour the egg mixture into the skillet with the vegetables. Add the black beans, cumin, salt, and pepper. Cook, stirring gently, until the eggs are fully cooked, about 5-7 minutes.
Remove the skillet from heat and stir in the shredded cheddar cheese until melted.
Warm the tortillas in a separate skillet or microwave for about 15 seconds until pliable.
Divide the egg mixture evenly among the tortillas, placing it in the center of each.
Fold the sides of the tortilla over the filling, then roll it up from the bottom to create a burrito.
Serve immediately with salsa and avocado slices if desired.