If grilling the corn, coat the grill grate with cooking spray and prepare a medium-high grill. Brush the corn with oil and place it on the grill. Cook, turning, until the corn is evenly cooked and browned in spots, 5 to 7 minutes. If boiling the corn, bring a large pot of water to a boil. Add the corn, cover, and remove from the heat. Let stand 3 to 5 minutes. Drain well.

When the corn is cool enough to handle, cut it from the cob and set aside.

If using the tortillas, line a baking sheet or large plate with paper towels. In a large skillet, heat 1 cup of the vegetable oil over medium-high heat until shimmery and hot but not smoking. Add a handful or two of tortilla strips-they can be touching but not overlapping-and fry just until lightly browned around the edges, about 45 seconds. Transfer the strips to the paper towel-lined baking sheet, tossing and moving them around so they take on squiggly shapes as they cool and harden. Continue in batches until all the tortilla strips are fried, adding more oil to the pan if necessary. Set aside.

In a small, preferably ovenproof, skillet, heat 1 teaspoon of the oil over medium heat. Add the onion and jalapeno and cook, stirring occasionally, until softened and fragrant, 2 to 3 minutes. Add the corn and tortillas, if using, and toss until warmed through. Add a pinch each of salt and pepper, or to taste. If necessary, transfer to a small, ovenproof dish.

Drizzle the aeoli on top and sprinkle with the cheese.

Place an oven rack 6 inches from the broiler and turn the broiler to high. Place the skillet or dish under the broiler and cook until browned and bubbling, about 5 minutes. Watch carefully so it does not burn. Serve.

In the work bowl of a food processor, place the egg yolk, mustard, lemon juice, and garlic. Process until blended. With the food processor running, add the oil just a few drops at a time. When the mixture begins to resemble mayonnaise (this will take a while because you're adding the oil so slowly), pour in the remaining oil in a very fine, slow stream.

If making the aioli to be drizzled over Vera Cruz corn or a salad, thin it if necessary by adding water 1 tablespoon at a time until it is pourable. If using as a condiment to be spread or dolloped, it may not need any thinning.

Add the chipotle and process until well blended. Add several pinches of salt to taste. Use at once or store in a tightly covered container in the refrigerator for 2 to 3 days.