Place a large heavy pot or Dutch oven over medium heat and add half of the oil. Add half of the turkey, season with salt and pepper and cook, stirring to break up the meat, until browned. Transfer to a bowl with a slotted spoon. Add the remaining oil and repeat with remaining turkey.
In a small saucepan, melt the butter over medium heat and stir in the hot sauce. Add the hot sauce mixture to the turkey, along with broth, tomatoes and juice and garlic. Bring the mixture to a boil over high heat, then reduce heat to low and simmer gently until reduced by 1/3, about 1 1/2 hours.
Stir in the green chiles with juice, tomato paste and pepper flakes and simmer 15 minutes more. Serve with cheese sprinkled on top.