In a skillet over medium heat, cook the breakfast sausage links until they're browned and cooked through, about 5-7 minutes. Drain any excess fat for a lighter filling.

In a bowl, whisk together large eggs, milk, salt, and pepper. Pour this mixture into the skillet and cook in butter until the eggs are fluffy and just set, about 3-4 minutes.

Unroll the refrigerated crescent roll dough onto a flat surface. Place a generous scoop of the sausage and egg mixture in the center of each triangle. Roll them up tightly, ensuring the filling is well enclosed.

Arrange the roll-ups seam side down on a baking sheet lined with parchment paper. Bake in a preheated oven at 375°F for about 12-15 minutes, or until golden brown and flaky.