Falafel: sort and wash chickpeas, and place in a large bowl. Cover with water to 2 inches above beans; cover and let stand 8 hours or overnight. Drain.
Preheat oven to 350º.
Combine chickpeas and next 11 ingredients (through 2 chopped garlic cloves) in a food processor; pulse 8 to 10 times or until finely chopped. Spoon mixture into a bowl. Add egg whites to chickpea mixture, and stir well. Let stand 15 minutes. Divide mixture into 16 equal portions, shaping each into a 1/2-inch-thick patty. Place patties on a baking sheet coated with cooking spray. Bake at 350º for 10 minutes or until lightly browned.
Relish: combine tomato and next 5 ingredients (through chile).
Sauce: combine yogurt, goat cheese, 1/8 teaspoon salt, and 1 minced garlic clove, stirring with a whisk until smooth.
Place 2 falafel patties in each pita half; spoon about 1/3 cup relish and 2 1/2 tablespoons sauce into each pita half. Serve immediately.