Heat a large skillet over medium-high heat and add the frozen diced potatoes with butter and the seasonings. Stir to coat, then cook undisturbed for 5-6 minutes until crisp on one side. Flip and continue cooking until golden brown. Then, transfer to a plate.
Add the sausage links to the same skillet and cook, turning occasionally, until nicely browned on all sides. While the sausage cooks, whisk the eggs in a bowl.
Remove the sausage from the pan and reduce the heat to medium, then add the eggs and cook gently. Slice the sausage into thin rounds while the eggs cook.
Stir the cheese into the eggs just before they finish cooking. Return the potatoes and sausage to the skillet and stir to combine.
Remove the egg mixture from the heat and fill the tortillas, rolling them into burritos. Butter the outside of each burrito and return them to the skillet over medium heat, cooking until crisp on both sides