Cook the sausage in a large skillet over medium-high heat until browned, 6–8 minutes. Drain on a paper towel-lined plate.
Leave about 1 tablespoon of fat in the pan, adding oil if needed. Add the potatoes, garlic powder, smoked paprika, and a pinch of salt. Cook 8–10 minutes, flipping occasionally, until crisp.
Whisk the eggs, milk, salt, and pepper.
Wipe the skillet, melt the butter over medium-low heat, and gently scramble the eggs until just set but still glossy.
Combine the eggs, potatoes, sausage, and green onions. Spread on a sheet pan and cool 10–15 minutes.
Warm the tortillas. Add about ¾ cup filling to each, top with cheese and salsa, fold in the sides, and roll tightly.
Toast seam-side down in a dry skillet over medium heat, 1–2 minutes per side.
Serve immediately, or cool, wrap individually, and freeze.