Preheat your oven to 400 degrees. Line a rimmed baking sheet with parchment paper.
Drizzle the cut side of the squash with about 1 tablespoon of olive oil. Sprinkle with 1/4 tsp. each salt and black pepper. Then place the squash cut-side-down on your parchment paper-lined baking sheet.
Cut the top 1/4-inch off the head of garlic, exposing some of the cloves inside. Place the head of garlic cut side-up on a square of aluminum foil. Drizzle about 1/2 tablespoon olive oil over the exposed cloves. Then fold up the sides of the aluminum foil to completely wrap up the garlic. Add this to your baking sheet with the squash.
Meanwhile, add the chiles to a heat-safe bowl and pour boiling water over them just until they’re covered. Set aside to let them rehydrate for about 30 minutes.
Fit the bowl of your food processor with the blade attachment. Scoop out the flesh from the squash and measure it so you have roughly 1 1/2 cups. Add this to your food processor. Unwrap the head of garlic and squeeze out 6-8 of the cloves and add them to your food processor.