Turkey Sloppy Joes

Ingredients:
2 tbsp butter
1 onion
salt and pepper to taste
1 1/4 pounds ground turkey plus 1/2 cup cold water
3/4 cup ketchup
1 1/2 tbsp brown sugar, or to taste
cayenne to taste
1/2 tsp Worcestershire sauce
1/2 tsp unsweetened cocoa
1 1/2 cup water, or as needed
1/3 cup chopped green onions
1/2 cup shredded white cheddar cheese