Combine the hatch red chiles, onion, garlic, lime juice, and cilantro in a blender.
Blend until a salsa consistency is created, smooth but with a little texture.
Season with salt to taste.
Preheat the oven to 450°F. Drizzle a thin coating of oil onto two sheet pans. Distribute 6 tortillas onto each pan, dredging each side in the oil. Season with a pinch of salt. Bake for 12 minutes, or until crisp.
Top the crispy tortillas with cheese. Bake for another minute or so until melted.
Build the tostadas by layering on the carnitas, salsa, sour cream, radish, and cilantro. Serve with lime wedges alongside.