Preheat oven to 350 degrees. Lightly brush tortillas with 4 tablespoons oil. Sprinkle with salt and pepper.
Press one of the tortillas into the ovenproof bowl. Bake 13 minutes; an adult should transfer shell to a rack to cool. Repeat with remaining tortillas.
Meanwhile, make beef filling: In a mixing bowl, combine beef with tomato paste, chili powder, cumin, and onion; season with salt and pepper. An adult should heat 2 tablespoons oil in a large skillet over medium heat, add beef mixture, and cook, breaking up meat with a spoon, 10 minutes. Transfer to a bowl; set aside.
Make dressing: In a jar, shake together vinegar, 1/2 cup oil, lime juice, and salt and pepper.
Using a box grater, shred cheese and carrots. An adult can prepare the other toppings, including corn: Heat 2 tablespoons oil in a medium skillet over medium high heat; add 10-ounce package frozen corn. Saute 3 minutes.
Divide beef filling among shells; add toppings and dressing.