Position a rack 5 to 6 inches from the heat source and preheat the broiler.
Put the tortillas in a food processor and pulse until they reach a fine crumb. Sprinkle half the tortilla crumbs evenly in a 1 1/2-quart baking dish. Arrange the tomatoes, cut-side up, over the crumbs. Sprinkle evenly with the garlic, oregano, 1/2 teaspoon salt and a couple turns of pepper. Top with the Cheddar.
Toss the remaining crumbs with the oil in a small bowl; sprinkle over the Cheddar. Broil until the Cheddar is melted and the crumbs are golden, about 10 minutes.