Whisk together eggs, milk, salt, and pepper in a bowl until well combined.
Melt butter in a large skillet over medium heat. Pour in egg mixture and cook, stirring gently, until eggs are scrambled and just set, about 5 minutes.
Warm tortillas in the microwave for 20-30 seconds or on a dry skillet to make them more pliable.
Divide scrambled eggs evenly among the four tortillas, placing them in the center.
Top each with cooked sausage, crispy hash browns, shredded cheese, salsa, and sour cream if using.
Fold in the sides of each tortilla, then roll up tightly from the bottom to enclose the filling.
Optional: Place burritos seam-side down on a hot skillet for 1-2 minutes to toast the outside and seal them shut.
Serve immediately with extra salsa on the side.