Melt butter in a heavy skillet over medium heat. Add onion; cook and stir until onion starts to brown, about 5 minutes. Season with salt and black.
Mix ground turkey and 1/2 cup cold water into onions. Cook and stir, breaking up the meat, until it begins to brown, about 2 minutes. Add ketchup, brown sugar, cayenne pepper, salt, and black pepper. Cook and stir until liquid is reduced, about 5 minutes. Stir in Worcestershire sauce and cocoa powder. Cook, stirring frequently and adding up to 1 cup water if needed, until meat is cooked through and liquid is reduced and thick, 20 to 30 minutes.
Preheat the oven's broiler and set the oven rack about 6 inches from the heat source. Toast hamburger buns until golden, 2 to 3 minutes.
Remove turkey mixture from the heat and mix in green onions and Cheddar cheese. Serve on toasted hamburger buns.
