Preheat oven to 375° and line a large baking sheet with parchment paper. Spoon about 2 tablespoons cheddar a few inches apart. Bake until bubbly and edges are beginning to turn golden, about 6 minutes. Let cool on baking sheet for a minute.
Meanwhile, grease bottom of a muffin tin with cooking spray, then carefully pick up melted cheese slices and place on bottom of muffin tin. Fit with another inverted muffin tin and let cool 10 minutes. If you don't have a second muffin tin, use your hands to help mold the cheese around the inverted tin.
In a large skillet over medium heat, heat oil. Add onion and cook, stirring occasionally, until softened, about 5 minutes. Stir in garlic, then add ground beef, breaking up meat with a wooden spoon. Cook until beef is no longer pink, about 6 minutes, then drain fat.
Return meat to skillet and season with chili powder, cumin, paprika, salt, and pepper.
Transfer cheese cups to a serving platter. Fill with cooked ground beef and top with sour cream, avocado, cilantro, and tomatoes.
For an easier time peeling the cheese from the parchment, cut the parchment paper into a 4 "box so that each of the cheese disks is in each of the parchment pieces.