In a large skillet over medium heat, brown the breakfast sausage links for 8-10 minutes, turning occasionally until they are golden brown and cooked through. Use a meat thermometer to ensure they reach an internal temperature of 160°F (71°C). Once fully cooked, remove them from the skillet and let them drain on paper towels to eliminate excess grease.

In a mixing bowl, whisk together 4 large eggs, 2 tablespoons of milk, and a pinch of salt and pepper until the mixture is pale and fluffy. Heat the same skillet over medium-low heat and pour in the egg mixture, stirring gently with a spatula. Cook for 4-5 minutes, or until the eggs are just set but still soft and creamy. Remove from heat and set aside.

Preheat your oven to 375°F (190°C) and lightly grease a baking dish with butter. Unroll the refrigerated crescent roll dough on a clean surface. Break the dough into individual triangles as indicated on the package.

Place one sausage link in the center of each dough triangle, followed by a generous spoonful of scrambled eggs. Sprinkle shredded cheddar cheese on top of the filling for extra flavor. Carefully roll up the dough from the wide end to the point of the triangle, ensuring the filling is enclosed snugly within the pastry.

Arrange the assembled roll-ups in the greased baking dish, spacing them apart to allow for expansion. Bake in the preheated oven for 12-15 minutes, or until the crescent rolls are golden brown and flaky.

Once baked, remove the roll-ups from the oven and let them cool for a few minutes. Serve warm and enjoy.