Cook the tater tot rounds according to the package instructions. Set aside.
Meanwhile, cook and crumble the sausage in a large nonstick skillet over medium-high heat until brown, about 10 minutes. Remove and drain on a paper towel-lined plate. Return the skillet to the heat.
Whisk together the milk and eggs in a large pitcher. Add the egg mixture to the skillet and scramble, about 5 minutes. Set aside.
To build the wraps, lay out the tortillas. Divide the sausage, eggs, Cheddar, Monterey Jack and tater tot rounds evenly and place on the very center of the tortillas. Top each with salsa, sour cream and avocado. Fold the edges of each tortilla up and over, creasing as needed, to the center and as tight as possible, so that you form a parcel with each one.
Heat 1 tablespoon of the olive oil in a large skillet over medium heat. Place one wrap in the skillet, seam-side down. Cook until golden, about 2 minutes, then flip and cook the other side for another minute or two. Repeat with the remaining tortillas and olive oil. Serve hot.