Melt the butter in a small saucepan over medium heat. Add the onion and cook until softened, about 5 minutes. Add garlic and stir briefly, about 30 seconds.
Add cherries, tomato sauce, sugar, molasses, apple cider vinegar, Dijon mustard, chili powder, pepper, salt, and cayenne pepper to the pot and stir to combine.
Bring to a boil over medium-high heat, and then reduce the heat to low. Simmer, stirring occasionally until the cherries are soft and the sauce thickens, about 30 minutes.
Puree the cherry barbecue sauce with an immersion blender or food processor. Use immediately, pour into a jar and refrigerate up to 2 weeks, or place in freezer containers and freeze up to a year. Makes about 2 cups.