Cut about 1/2 inch off the top of the garlic. Cut the roots so that the garlic will sit flat. Remove the papery skin from the garlic, but leave enough so that the cloves stay together. Put garlic into small casserole dish or baking pan, drizzle olive oil over it, and cover with foil. Bake in a preheated 325 degree oven for 1 hour. Remove the garlic and let it cool until you can handle it.
Combine water, pineapple juice, teriyaki sauce, soy sauce, brown sugar, & cayenne pepper in a medium saucepan over medium high heat. Stir occasionally until mixture boils. Reduce heat until mixture is just simmering.
Squeeze the sides of the heat of garlic until the roasted garlic is squeezed out. Measure 2 tsp of garlic and whisk to combine. Add remaining ingredients to the pan and stir.
Let mixture simmer for 50 minutes, or until sauce has reduced by half and is thick and syrupy. Make sure it doesn’t boil over.
Roasting a whole head of garlic at 325 degrees for one full hour transforms pungent alliums into a sweet, paste-like consistency that dissolves completely into liquid without leaving behind fibrous chunks. Skeptics often assume copycat restaurant reductions will crystallize into rock-hard caramel upon cooling, but the high proportion of pineapple juice and water keeps the 1 1/3 cups of dark brown sugar fluid. When bubbling this mixture in a medium saucepan over medium-high heat, watch for the bubbles to transition from rapid, watery pops to slow, heavy craters that indicate the water content has successfully evaporated. The reduction requires roughly 50 minutes of simmering to reach a thick, syrupy coat that clings to the back of a spoon. Leftovers stored in a sealed glass jar will thicken further in the refrigerator due to sugar retrogradation, but gentle reheating on the stovetop with a single teaspoon of water restores the original glossy viscosity immediately. Using freshly squeezed lemon juice imparts a sharp citrus brightness that bottled alternatives lack, cutting through the heavy sweetness of the soy sauce and crushed pineapple base.