Heat the oven to 450°F with a metal sheet pan inside. Brown the beef in a skillet over medium-high heat for 5 to 6 minutes, breaking it into fine crumbles and cooking it to 160°F.
Stir in cumin, chili powder, salt and 2 tablespoons salsa. Cook for 1 minute until nearly dry, then let cool briefly while you warm the tortillas.
Wrap the tortillas in a clean damp towel and microwave for 45 to 60 seconds until very flexible. Divide the beef and cheese in narrow lines across them and roll tightly.
Carefully brush the hot sheet pan with half the oil. Arrange the taquitos seam-side down and brush with the remaining oil; bake for 8 to 10 minutes, until crisp, golden at the edges and 165°F in the filling.
Serve three per person with shredded lettuce, sour cream and the remaining salsa.