Combine the aioli ingredients in a small dish and whisk to combine. Set aside.
For The Breakfast Burritos
Cook the bacon or sausage, or a combination of the two until the bacon is crisp and the sausage is crumbled and no longer pink. Drain any excess fat. Set aside.
Whisk the eggs together until there are no dark yellow streaks of yolk visible. Heat the butter in a skillet over medium heat. Add the eggs and stir slowly, until the eggs come together and become cooked. Remove from heat when they are still slightly shiny. Season with salt and pepper.
Warm the tortillas so they are pliable. Divide the ingredients evenly between the 6 tortillas, placing the filling in the center of each. Add 1 to 2 tbsp of the aioli to each burrito. Fold over one end of the tortillas, covering the filling, then pull up the sides of the tortilla, while rolling and securing the filling inside.
Serve the burritos immediately or wrap in foil and store in a zip-close bag in the freezer for up to 1 month.
Notes
REHEAT BURRITOS FROM FROZEN: Unwrap foil from burrito and wrap in a damp paper towel. Place in microwave and heat for 3 to 4 minutes, flipping over halfway, until heated through.