In a medium saucepan, heat olive oil over medium heat. Add the chopped onion and garlic, stirring frequently, until the onions are translucent. Add the remaining ingredients except horseradish (apple juice, tomato paste, vinegar, brown sugar, molasses, yellow mustard, paprika Worcestershire sauce, salt, and pepper) and bring to a simmer, stirring frequently. Turn heat to low, cover, and cook for 30 minutes or until the sauce reaches the desired consistency. Add the horseradish and serve as desired.