1. Preheat oven to 450°.
2. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, 3/4 teaspoon salt, and baking powder in a food processor; pulse to combine. Combine 1/3 cup oil and 5 tablespoons water. With processor on, slowly add oil mixture through food chute; process until dough is crumbly. Turn dough out onto a lightly floured surface. Knead 1 minute; add additional flour, if necessary, to prevent dough from sticking. Gently press dough into a 5-inch disk; wrap in plastic wrap, and chill at least 30 minutes.
3. Arrange tomatillos, jalapeños, and 4 garlic cloves in a single layer on a baking sheet coated with cooking spray; lightly coat vegetables with cooking spray. Bake at 450° for 25 minutes or until charred. Combine tomatillo mixture, juice, 1/4 teaspoon salt, 1/4 teaspoon black pepper, and sugar in a blender; pulse until finely chopped, scraping sides.
4. Reduce oven temperature to 400°.
5. Heat a large nonstick skillet over medium heat. Add remaining 1 teaspoon oil to pan; swirl to coat. Add onion; sauté 5 minutes or until tender. Chop remaining 2 garlic cloves. Add garlic to pan; sauté 1 minute. Add chicken and mushrooms; cook 6 minutes or until chicken is browned, stirring to crumble. Cool slightly. Add 1/2 cup tomatillo sauce, remaining 1/4 teaspoon salt, remaining 1/4 teaspoon black pepper, and cheeses, stirring well to combine.
6. Combine egg and 1 teaspoon water in a small bowl, stirring well with a whisk. Remove dough from refrigerator; let stand 5 minutes. Divide dough into 8 equal portions, shaping each into a ball. Roll each dough portion into a (6-inch) circle on a lightly floured surface. Working with 1 portion at a time (cover remaining dough to keep from drying), spoon 3 tablespoons chicken mixture into center of each circle. Moisten edges of dough with egg mixture; fold dough over filling. Press edges together with a fork to seal. Brush remaining egg mixture evenly over empanadas. Place empanadas on a parchment-lined baking sheet coated with cooking spray. Bake at 400° for 24 minutes or until lightly browned. Serve immediately with remaining tomatillo sauce.
MyRecipes is working with Let's Move!, the Partnership for a Healthier America, and USDA's MyPlate to give anyone looking for healthier options access to a trove of recipes that will help them create healthy, tasty plates. For more information about creating a healthy plate, visit www.choosemyplate.gov.