To prepare patties, place pinto beans in a medium bowl; partially mash with a fork. Add Monterey Jack cheese and next 5 ingredients (through egg white); stir until well combined. Form into 4 (1/2-inch-thick) oval patties.
Heat canola oil in a large nonstick skillet over medium-high heat. Add patties; cook 3 minutes on each side or until patties are browned and thoroughly heated.
To prepare spread, while patties cook, combine the avocado, plum tomato, red onion, sour cream, lime juice, and salt. Place 1 patty in each pita half. Spread about 2 tablespoons avocado spread over patty in each pita half.