Typically cactus paddles are sold with the thorns removed. If not, use a small paring knife or vegetable peeler to remove and discard thorns and any blemishes on cactus. Wash and use paper towels to pat the cactus paddles dry. Rub both sides with vegetable oil and sprinkle 1/4 tsp of salt and 1/4 tsp of pepper over all. Set aside on platter while grill is heated to 400°F.
In a salad bowl, whisk together the lime juice, olive oil, remaining salt and pepper and cumin. Add the minced chiles and garlic. Allow to rest while grilling the cactus.
Transfer cactus paddles to the hot grill. Cook each side for 4 minutes or until grill marks are visible. Remove from heat and allow to cool while you assemble the remaining salad ingredients.
Add the tomatoes, red onion, avocado and cilantro to the marinated chile dressing. Toss to coat.
Slice each cactus in half lengthwise. Stack both pieces and slice into 1/2 inch strips. Repeat with remaining paddles. Add to the salad and toss to mix. Serve immediately or chill for an hour.
Sprinkle the cotija cheese on top before plating. Serve as a starter salad, with tortilla chips as an appetizer or with grilled grilled meats.