Preheat the oven to 400°F and line a baking sheet with parchment paper.
Add the chickpeas, parsley, cilantro, garlic, onion, cumin, salt, pepper, and flour to a food processor. Pulse until the mixture holds together when pressed but still has some texture.
Scoop the mixture into 12 portions and shape each into a patty. Place the patties on the prepared baking sheet and brush both sides with olive oil.
Bake for 20 minutes, flipping halfway through, until the falafel are golden and firm.
While the falafel bake, whisk the tahini, lemon juice, and water in a small bowl until smooth. Add a little extra water if needed to reach a pourable consistency.
Warm the pita breads in the oven for the last 3 minutes of baking time.
Divide the mixed greens, cherry tomatoes, and cucumber among four bowls. Top each with three falafel patties, drizzle with tahini sauce, and serve with warm pita on the side.