Heat oven to 450° F. Line a rimmed baking sheet with foil. Layer the tortilla chips, Cheddar, and onion on the prepared pan. Bake until the cheese is melted, 4 to 5 minutes.
Meanwhile, heat the oil in a large skillet over high heat. Add the shrimp, cayenne, and ½ teaspoon salt. Cook, tossing often, until the shrimp are opaque, 2 to 3 minutes.
Serve the nachos topped with the cooked shrimp, avocados, radishes, cilantro, and sour cream.