Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Combine turkey, onion, chilies, cilantro, garlic, jalapeno pepper, cumin, oregano, cinnamon, salt, and pepper in a large bowl; gently mix until combined. Form into four 1/4-pound burgers.
Cook on the preheated grill until cooked through, about 6 minutes per side. An instant-read thermometer inserted into the center of the burgers should read at least 165 degrees F (74 degrees C).
Remove from the heat, add Monterey Jack cheese slices, and tent with aluminum foil until cheese is melted, 2 to 3 minutes.