To prepare patties, place pinto beans in a medium bowl; partially mash with a fork. Add cheese and next 5 ingredients (through egg white); stir until well combined. Shape bean mixture into 4 (1/2-inch-thick) oval patties.
Heat oil in a large nonstick skillet over medium-high heat. Add patties to pan; cook 3 minutes on each side or until patties are browned and thoroughly heated.
To prepare spread, combine avocado and next 5 ingredients (through salt), stirring well. Place 1 patty in each pita half. Spread about 2 tablespoons avocado spread over patty in each pita half; top with onions and greens.