Heat the oil over medium heat in a large, wide skillet. Working in batches, add the hashbrowns in a thin layer, dust with spices, and press gently with a spatula. Cook until golden and crispy, then flip and repeat. Remove in chunks to a paper-towel-lined plate and season with salt.
Brown the ground sausage in a large nonstick skillet. Drain off excess grease and turn the heat down.
Add the beaten eggs to the pan with the sausage and gently stir until just barely set.
Lay one large tortilla flat and cut it into fourths to use as filler pieces.
Layer hashbrowns, the sausage-and-egg mixture, and cheese onto each remaining tortilla. Place a filler piece over the center.
Fold the outer edges of the tortilla inward, one section at a time, until it forms a star shape.
Place the wrap seam-side down in a hot, oiled skillet. Cook a few minutes on each side, until firm, crunchy, and golden brown. Cut and serve.