Preheat the oven to 350°F (175°C).
In a medium bowl, whisk together the eggs, whole milk, salt, and black pepper until well combined.
Heat a large non-stick skillet over medium heat and add the breakfast sausage. Cook, breaking it up with a spoon, until browned and cooked through, about 5-7 minutes. Remove the sausage from the skillet and set aside.
In the same skillet, add the butter and let it melt over medium heat.
Pour the egg mixture into the skillet and cook, gently stirring, until the eggs are softly scrambled and just set, about 3-4 minutes. Remove from heat.
Lay out the flour tortillas on a clean surface. Divide the scrambled eggs evenly among the tortillas, placing them in the center.
Top the eggs with the cooked sausage and then sprinkle shredded cheddar cheese over the sausage.
Carefully roll up each tortilla, folding in the sides as you go to enclose the filling.
Place the roll-ups seam side down in a baking dish.
Bake in the preheated oven for 10 minutes, or until the tortillas are lightly golden and the cheese is melted.
Remove from the oven and let cool for 2 minutes before serving.