Place the oven rack in the center position and preheat the oven to 400°F. Peel away the outer skin of the head of garlic while leaving all of the cloves intact. Cut a 1/2-inch slice off the top of the head of garlic so that the cloves are exposed; discard the top. Set the head of garlic on a square of foil, drizzle with 1 tablespoon extra-virgin olive oil, and wrap in the foil. Set on a pan and roast for 40 minutes. Carefully open the top of the foil packet (watch for hot steam!) and use a sharp knife to pierce down into a clove; the garlic should be completely softened and golden brown. If it's not, re-wrap the foil and continue to roast, checking every 5 minutes or so, until garlic is done. Unwrap the garlic and set aside to cool.
Once the garlic is cool enough to handle, squeeze the resulting garlic paste into the bowl of a high-powered blender or large food processor. Add the drained chickpeas, tahini, olive oil, lemon juice, salt, and cayenne, plus 3 tablespoons of the reserved chickpea liquid. Puree until smooth, adding additional chickpea liquid 1 tablespoon at a time until a desirable consistency is reached. Continue to puree for another minute or two until completely smooth, scraping down the sides as needed and seasoning with additional salt to taste, if necessary. At this point, you may discard any leftover reserved chickpea liquid.
Transfer the hummus to a serving dish and swirl the surface. Drizzle the top with extra-virgin olive oil and sprinkle with smoked paprika and minced parsley. Serve with pita wedges or chips, crackers, and/or raw vegetable dippers.
When I opened my jar of tahini, it was extremely separated and stirring it with a spoon wasn't going to be effective. So I scraped the whole mess into my food processor before beginning the recipe, pureed until the tahini was smooth and incorporated, and then transferred all of it back into the tahini jar. At that point, I proceeded with my recipe (without cleaning out the food processor first), measuring 1/4 cup tahini from the jar back into the food processor when the time came.