Heat the grill to medium-high--about 400 degrees. (OR cook on a non-stick grill pan on medium -igh heat on the stove top, OR broil on high heat on a non-stick sheet pan.)
To make the marinade/sauce, add the dry ingredients-garlic, cayenne, chopped ginger, thyme, jalapeno, green onions (and Jerk seasoning, if desired) to a food processor. Process until finely chopped and blended. Add liquid ingredients (BBQ sauce, lime juice, and soy sauce) and process for about 30 seconds or until the ingredients are thoroughly blended into a slightly chucky sauce.
Divide the marinade/sauce in half. Brush half of the marinade on each side of the eggplant, and reserve the rest as a sauce for serving.
Cook the eggplant (on the grill, stove top or under the broiler) about 5 minutes per side, or until browned and tender.
To reduce the sodium in this recipe, use low sodium tomato paste in the BBQ sauce or buy low sodium BBQ sauce. This recipe works well with either homemade or store-bought vegan BBQ sauce, like Amy's (brand) Sweet & Spicy.
Jerk seasoning is readily available at most stores, or this simple version works well with this recipe: I teaspoon cinnamon, ½ teaspoon allspice, ¼ teaspoon each of ancho chili powder, sea salt and freshly ground black pepper, ½ teaspoon sweetener like erythritol (Swerve), maple syrup or date paste.