In a skillet over medium heat, add the fully cooked breakfast sausage links and cook for about 5-7 minutes, turning occasionally until browned.
In the same skillet, reduce the heat to low and add 1 tablespoon of butter. Whisk together the eggs, milk, salt, and pepper, then pour into the skillet to scramble for 3-4 minutes.
Preheat your oven to 375°F (190°C) while the eggs are cooking, and prepare a baking sheet with parchment paper.
Unroll the crescent roll dough on a floured surface, separate it into triangles, and press them out slightly larger.
Place one sausage link, a spoonful of scrambled eggs, and a handful of shredded cheese on each triangle, towards the wide end.
Roll each triangle from the wide end, tucking in the sides to seal the filling.
Bake the roll-ups on the prepared baking sheet for 12-15 minutes or until golden brown.
Allow cooling for a couple of minutes before serving, optionally brushing with melted butter.
Notes
Experiment with different cheeses or add diced vegetables to customize your roll-ups.