Cook the Beef: Heat 1 tablespoon olive oil in a large cast iron skillet over medium-high heat. Add ground beef and cook until browned, about 3–5 minutes, crumbling as it cooks. Stir in taco seasoning. Drain excess fat and set aside.
Prepare Tortillas & Cheese: Heat tortillas according to package instructions. Heat nacho cheese according to package instructions.
Assemble the Wraps: Working one at a time, place a portion of the beef mixture in the center of each tortilla. Top with nacho cheese and a tostada shell. Spread sour cream over the tostada, then layer lettuce, tomato, and shredded cheese.
Fold the Tortilla: Fold the edges of the tortilla up and over the filling, working around the wrap until it is tightly sealed.
Cook the Wraps: Heat the remaining 1 tablespoon olive oil in the skillet. Place the wrap seam-side down and cook until golden brown, about 2 minutes. Flip and cook the other side for 1–2 more minutes.