Preheat oven to 350℉.
Lay tortillas on a countertop and lightly brush on both sides with oil. Gently fit tortillas into the cups of a standard muffin tin, pleating them to fit. Bake until lightly browned around the edges, 10-15 minutes. Transfer tortilla cups to a rack to cool.
In a large bowl stir to combine cotija or Parmesan, mayonnaise, lime juice, and chili powder. Add corn, onion, and cilantro and stir to coat with dressing.
When tortilla cups are cool, fill with corn mixture and serve immediately.