Set the red pepper over a gas burner on high and rotate with tongs until blackened on all sides, about 5 minutes. Transfer the pepper to a bowl, cover with a dish towel, and set aside to steam for a couple of minutes. Rub the pepper with paper towels to remove the skin¿don't worry if bits of skin remain; they'll lend some charred flavor. Stem and seed the pepper, and then roughly chop. (If using a jarred pepper, simply stem, seed and chop.)
In a food processor, combine the chickpeas, tahini, basil, garlic, lemon juice and chili powder. Process until combined. With the motor running, drizzle in oil and then slowly add water (about 1/4 cup) until the desired consistency is reached; bear in mind that the hummus will thicken in the refrigerator. Season with salt and pepper, and chill before serving to let the flavors meld.
Garnish the hummus with a basil sprig, and drizzle with olive oil. Serve with pita chips and crudites.