Heat the oil in a medium skillet over medium heat. Add the onion and cook until translucent, about 3 minutes. Add the ketchup. (If using a 20-ounce bottle pour the cider in the emptied bottle and shake to get every last little bit of ketchup.) Add the cider, honey, lemon juice, Worcestershire, Dijon mustard, and cayenne pepper. Season with salt and pepper.

Bring to a boil, decrease the heat to low, and simmer until the flavors have smoothed and mellowed, at least 10 and up to 30 minutes. Taste and adjust for seasoning with salt and pepper. Store in an airtight container in the refrigerator up to 5 days.