1. To prepare Tahini-Yogurt Sauce, combine first 6 ingredients in a small bowl. Cover and chill until ready to serve.
2. To prepare falafel, combine 1 1/3 cups boiling water and bulgur in a small bowl. Cover and let stand 25 to 30 minutes or until tender. Drain.
3. Drop garlic through food chute with processor on; process until minced. Add bulgur, parsley, and next 6 ingredients (through egg white); process until smooth. Divide mixture into 8 equal portions, shaping each into a 1/2-inch-thick patty. Place patties on a baking sheet; cover and chill 30 minutes.
4. Heat 1 1/2 tablespoons oil in a large nonstick skillet over medium-high heat. Add 4 patties; cook 3 minutes on each side or until golden brown. Repeat procedure with remaining 1 1/2 tablespoons oil and 4 patties.
5. Spread 1 tablespoon Tahini-Yogurt Sauce inside each pita. Fill each pita half with 2 patties. Divide tomato, cucumber, and red onion evenly among pita halves, and drizzle evenly with 1 tablespoon sauce.