Cook the sausage in a skillet over medium heat for 7 minutes, breaking it apart, until browned and 160°F, then transfer it to a bowl.
Add the oil and potatoes, season with salt and black pepper, cover and cook for 8 minutes.
Uncover and cook the potatoes for 5 more minutes until tender with crisp brown edges.
Add the eggs and stir for 3 minutes until set to 160°F, then fold in the sausage.
Cool the filling for 2 minutes and divide it among the tortillas with the cheddar.
Fold in the sides, roll tightly and toast seam-side down for 2 minutes.