Place chiles on a preheated grill (or use a broiler with the oven rack very close to the heat) and turn peppers occasionally until blackened all over and tender.
Wrap the peppers in plastic wrap and let stand 15 minutes. Cut peppers in half, then use paper towels to rub away skin and seeds. Slice peppers into strips and set aside.
Divide ground beef into 4 equal pieces. Form into patties and season outside with salt and pepper.
Meanwhile, toss shiitake mushrooms with olive oil and salt and pepper taste. Using a grill basket or a sheet of tin foil, grill shiitake mushrooms, turning occasionally until mushrooms are tender. Keep warm.
Grill burger patties over preheated grill, flipping once until burgers register your desired degree of doneness (140-145F for medium, 150-155 for medium-well) Top burgers with cheese, cover, and let stand 30-60 seconds until cheese is melted.
Place burger patties on toasted buns and top with shiitake mushrooms, sliced chiles, bacon, pickled radishes, and cilantro. Serve immediately and enjoy!