Place beans in a large container and cover by 3 to 4 inches with water; they will triple in size. Let soak overnight, add more water if necessary.
Drain beans, saving the soaking water, and place in a food processor fitted with the blade attachment. Add remaining ingredients except oil and pulse until all ingredients are minced and mixture comes together, but doesn’t form a paste. Stop motor occasionally to scrape down sides of bowl and add soaking water a tablespoon at a time if necessary to get mixture moving. Shape into 2-inch patties and set on a baking sheet. (These can be made a couple of hours in advance, covered and chilled).
Pour oil in a large cast iron skillet or frying pan until it reaches an inch up the sides of the pan. Heat oil over medium heat until it reaches 350°F on a deep fat fry thermometer or until a wooden chopstick inserted in the oil is immediately covered with bubbles.
Add patties a few at a time and let fry until browned on one side, about 1 to 2 minutes. Flip and repeat on the other side. (If they are browning too quickly reduce heat or raise the heat if not fast enough.) Remove from oil to a paper towel-lined baking sheet and sprinkle with salt. Repeat with remaining patties.
You can serve the falafel on their own with the tahini sauce as an appetizer or make into a sandwich by tucking them into a toasted pita or lavash with some lettuce, cucumbers and olives and a drizzle of sauce on top.
For the Sauce:
Combine all ingredients in the bowl of a food processor fitted with the blade attachment. Process until smooth, adding more water if necessary to make into a pourable sauce.