In a medium bowl, whisk together the eggs, milk, salt, and black pepper until well combined.
In a large skillet over medium heat, melt 1 tablespoon of butter. Add the egg mixture and cook, stirring gently, until the eggs are scrambled and just set, about 3-4 minutes.
Remove the skillet from heat and stir in the crumbled sausage and shredded cheese until the cheese is melted and everything is evenly mixed.
Lay out the tortillas on a clean surface. Divide the egg and sausage mixture evenly among the tortillas, placing it in the center of each.
Roll up each tortilla tightly, folding in the sides as you go to secure the filling.
In the same skillet, melt the remaining tablespoon of butter over medium heat. Place the roll-ups seam-side down in the skillet and cook until golden brown, about 2-3 minutes per side.
Remove from the skillet and let cool slightly before slicing in half to serve.