In a food processor fitted with a metal blade, combine chickpeas, garlic, shallot, parsley, cumin, coriander, and flour and season with salt and pepper. Pulse until mixture is coarse and mealy—do not over blend!
Form mixture into falafel balls about 2" in diameter, squeezing to compact. Transfer to chill in the refrigerator while oil heats.
In a pot, heat 1” vegetable oil until a drop of water added to the oil sizzles and pops.
Fry falafels until golden, then transfer to a paper towel-lined plate and season immediately with salt.
To make yogurt sauce: In a medium bowl, whisk together yogurt, lemon juice, oil, and dill. Season with salt and pepper.
To make tahini sauce: In a medium bowl, whisk together tahini, garlic, lemon juice, and warm water. Season with salt. (If you prefer a thinner sauce, whisk in more warm water 1 tablespoon at a time.)
Serve falafels in pita with lettuce, tomatoes, and cucumber and drizzle with either sauce.