Place bulgur in a bowl and pour in boiling water to cover by 1 inch. Let stand until softened, 20 to 30 minutes.
Meanwhile, prepare Yogurt-Garlic Sauce.
Place bread in a food processor and process into fine crumbs. Transfer to a mixing bowl. With the motor running, drop garlic through the feed tube and process until minced. Add pine nuts and pulse until chopped. Add onion and pulse until chopped. Transfer to another bowl. Put mushrooms in the food processor and pulse until coarsely chopped.
Heat oil in a large nonstick skillet over medium-high heat. Add the onion mixture and saute until softened, about 3 minutes. Add the mushrooms and cook, stirring, until softened and most of their liquid has evaporated, 5 to 6 minutes more. Add the vegetables to the breadcrumbs.
Drain the bulgur and stir into the vegetable-breadcrumb mixture. Season with salt and pepper. Add egg and mix well.
Preheat broiler. Coat a baking sheet with cooking spray.
Shape the burger mixture into 6 patties and place on the prepared baking sheet. Broil until lightly browned, 5 to 7 minutes per side. Tuck the burgers into pitas along with lettuce, tomato and the yogurt sauce.
Note: Bulgur, a staple in the Middle East, is made from wheat kernels that have been steamed, dried and crushed. It can be simmered in broth or water for a fiber-rich pilaf, or it can simply be plumped in boiling water for use in salads or vegetarian patties.