Place the first 9 ingredients (chickpeas through egg) in a high speed food processor and pulse on high until the mixture is smooth and everything is evenly incorporated. (It's ok if it's slightly chunky - texture is good.)
Remove mixture from the food processor and place into a medium bowl. Working a couple tablespoons at a time, slowly stir in the bread crumbs. Continue adding breadcrumbs until the mixture is dry enough to form patties.
Divide the bowl into quarters with your hands and carefully form four patties. Place patties on a platter drizzled with a little olive oil and flip once so the patties are lightly oiled on both sides.
To cook, heat a non-stick griddle or frying pan over medium-high heat. Brown patties until golden brown, about 4-5 minutes per side.
While the patties are cooking, make the tzatziki sauce in a separate bowl. To serve, place one patty in between each hamburger bun and top with tzatziki sauce and other designed condiments.1