Season chicken with salt and pepper and wrap in parchment paper-lined foil; place on a rimmed baking sheet. Transfer to oven and bake until chicken is cooked through, about 1 hour.
Unwrap chicken; remove skin and discard. Remove chicken from bones and cut into small pieces; discard bones and set chicken aside.
Decrease oven temperature to 350 degrees. Layer one third of the chips on a large ovenproof platter or in a baking dish. Top with 1 cup cheddar and 1 cups of Monterey Jack. Repeat process twice with remaining chips and cheese. Transfer to oven and bake until cheese is melted, 5 to 10 minutes.
Remove from oven and top with reserved chicken, both salsas, cheese sauce, guacamole, sour cream, jalapenos, radishes, and cilantro; serve immediately.
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