Heat oil in a medium saucepan over medium heat. Add onion and cook until translucent, lowering the heat if it starts to brown. Add garlic and ginger; cook a minute more.
Stir in peaches and remaining ingredients, starting with the lesser amount of chipotle if you're concerned about heat - you can always add more. Simmer until peaches are tender, then transfer to a blender and carefully puree until smooth (work in two batches if your blender doesn't have steam vents in the lid).
Reserve 1/2 cup of sauce in a separate bowl for basting the chicken, and transfer the rest to a dish for serving at the table.
For Chicken & Rub:
Stir together paprika, cumin, salt, allspice and black pepper. Rub it into the entire surface of the chicken thighs, gently working it under the skin. Let stand for 30 minutes at room temperature before grilling.
To prepare grill, light one burner (or half of the grill) over medium heat and close the lid; preheat for 10 minutes. Grill chicken skin-side down over direct heat (over the lit burner) to crisp the skin, about 8 minutes. Watch closely for flare-ups, moving the chicken to the unlit side as needed until they subside. Move chicken over to indirect heat (unlit side) and cook with lid closed as much as possible for 15 minutes, flipping once.
Brush both sides of chicken with 1/2 cup of the BBQ sauce and continue cooking until meat reaches 165ºF on a thermometer, or until juices run clear when meat is pierced - 15-20 minutes more. Serve with more BBQ sauce on the side.
Any remaining sauce will keep for 2 weeks in the fridge.