Mexican Shredded Beef Breakfast Burritos

Name: Mexican Shredded Beef Breakfast Burritos
Ingredients: 2 to 2 1/2 pounds beef shoulder roast boneless arm chuck roast boneless, or boneless blade chuck roast; 1 large onion sliced; 2 tablespoons minced garlic; 1 1/2 cups salsa or picante sauce; For the breakfast burritos:; 1 teaspoon unsalted butter; 8 large eggs; 1 tablespoon milk; 1/4 teaspoon salt; 1/8 teaspoon pepper; 6 extra-large flour tortillas 10 inches wide; 12 ounces Mexican shredded beef; 6 ounces shredded cheddar cheese; Additional salsa or picante sauce for serving optional