Add onion and cook, stirring occasionally, until soft, about 10 minutes. Add a little water if the onion is sticking.
Add garlic and jalapeño pepper and cook, stirring, until fragrant, about 1 minute.
Add bourbon, increase heat to high and bring to a boil. Cook until most of the liquid has evaporated, about 5 minutes.
Stir in blueberries, ketchup, vinegar, brown sugar, molasses and allspice. Return to a boil. Reduce the heat and simmer, stirring occasionally, until thickened, about 30 minutes.
Let cool about 20 minutes. Transfer to a food processor, blender or Vitamix and blend until smooth.
Cover and refrigerate until ready to use – will keep for about 2 weeks.
Notes
* Freeze for up to 3 months. * Want to make this gluten-free? Use an gold rum in place of bourbon.