Place chickpeas in a large bowl and cover with water by several inches. Soak for 24 hours.

The next day, drain the chickpeas well and rinse in a colander, then shake dry.

Half at a time, pour chickpeas into a food processor fitted with the blade attachment.

Process, stopping to scrape down the sides as necessary, until chickpeas begin to form a very coarse paste. It should just hold together when you squeeze it in your palm but still be made up of thousands of little chickpea bits.

Scrape the first batch of chickpea paste into a large mixing bowl.

After pulsing the second batch, add the onion, garlic, cilantro, parsley, baking powder, coriander, cumin, cayenne, salt, and pepper to the food processor right on top of the pulverized chickpeas. Pulse until combined.

Add the second batch of chickpea mixture to the bowl and stir well but gently so that all ingredients are well-distributed.

Cover the bowl and refrigerate for an hour.

Using a medium cookie scoop or a heaping measuring tablespoon, scoop out portions of the mixture and use your hands to roll lightly into balls. The balls should stick together well, but try not to work the batter too much.

Divide falafel between two parchment-lined half-sheet pans, and refrigerate for 15 minutes.

Pour oil into a large, heavy pot to a depth of about 1 inch (I use a 5-½ quart Dutch oven).

Insert a candy/deep fry thermometer and heat oil over medium heat until it reaches 350°F.

Carefully drop balls one by one into the hot oil. Don't crowd them—you'll fry in batches of about 8 at a time, depending on the size of your pot.

Fry until golden brown on all sides, about 3 to 4 minutes.

Drain on a wire rack set over a baking sheet. Keep warm in a 250°F oven if you like.

Place the tahini paste into a medium mixing bowl.

Add the lemon juice and cumin.

Using a chef's knife, mince the garlic. Sprinkle the salt overtop, and use the side of the knife to scrape and smash the garlic and salt mixture into a paste. Add the paste to the bowl.

Stir well. Mixture will seize up a bit and become very thick.

Add the water a tablespoon at a time (you may need even more than indicated, depending on the thickness of the tahini paste), and stir well until sauce is homogenous and is the consistency you'd like.

Cut pitas in half and gently separate each half to make a pocket.

Into each half, spoon some hummus if using, and tuck a couple of pieces of falafel.

Tuck in lettuce, tomatoes, onion, and olives.

Drizzle with tahini sauce, sprinkle with lemon juice, and serve right away.