Cut about 1/2-inch from the top of the garlic head. Trim the roots so the garlic sits flat. Remove some of the outer layers of papery skin from the garlic, leaving enough so that the cloves stay together. Put garlic into a small casserole dish or baking dish, drizzle olive oil over it, and cover with a lid or aluminum foil.
Roast garlic in preheated oven until the cloves are soft, about 1 hour.
Stir brown sugar, pineapple juice, water, teriyaki sauce, and soy sauce together in a saucepan and bring to a boil; reduce heat to low and keep at simmer.
Squeeze sides of the garlic until the pasty roasted garlic emerges. Measure 2 teaspoons roasted garlic into the saucepan and incorporate into the sauce with a whisk. Reserve remaining roasted garlic for another use.
Stir lemon juice, white onion, bourbon whiskey, pineapple, and cayenne pepper into the sauce; bring to a simmer and cook until the volume of the liquid reduces by half and is thick and syrup-like, 40 to 50 minutes.
