To prepare chicken, combine first 5 ingredients in a large zip-top plastic bag; add chicken to bag. Seal and marinate in refregerator at least 4 hours or overnight.
To prepare crust, lightly spoon 1 cup flour into a dry measuring cup; level with a knife. Combine 1/4 cup flour, water, and vinegar in a small bowl; stir with a whisk to create a slurry. Set aside. Combine 3/4 cup flour, 1/2 teaspoon chili powder, 1/2 teaspoon cumin, and 1/4 teaspoon salt in a large bowl; cut in shortening with a pastry or 2 knives until mixture resembles coarse meal. Add slurry to flour mixture; toss with a fork until moist. Press mixture gently into a 6 x 4-inch rectangle on heavy-duty plastic wrap; cover with additional plastic wrap. Roll dough, still covered, into a 12 x 7-inch rectangle; freeze 10 minutes.
Preheat oven to 400º.
Remove chicken from bag; discard marinade. Heat 1/2 teaspoon oil in a large nonstick skillet over medium-high heat. Add chicken to pan; cook 5 minutes or until browned. Remove chicken from pan; place in a large bowl. Add corn to pan; sauté 6 minutes or until browned. Add 1/4 cup broth, scraping pan to loosen browned bits. Add the corn mixture to the chicken. Add 1/2 teaspoon oil to pan. Add zucchini; cook 6 minutes or until brown. Add zucchini and green chiles to corn mixture; toss well.
Lightly spoon 1/3 cup flour into a dry measuring cup; level with a knife. Combine flour and 1/2 cup broth in a small bowl; stir with a whisk. Place flour mixture in a saucepan; gradually add 1 1/2 cups broth, 3/4 teaspoon salt, 1/2 teaspoon chili powder, 1/4 teaspoon cumin, and black pepper. Bring to a boil over medium heat; cook 5 minutes or until thick, stirring constantly. Pour over chicken mixture; toss to coat. Spoon mixture into an 11 x 7-inch baking dish coated with cooking spray.
Remove 1 sheet of plastic wrap from dough; let stand 1 minute or until pliable. Fit dough over chicken mixture. Remove top sheet of plastic wrap, pressing dough to edge of dish. Cut 5 slits in crust to allow steam to escape. Bake at 400° for 35 minutes or until crust is golden brown and bubbly around the edges. Let stand on a wire rack 10 minutes.