COMBINE flour, salt and baking powder in medium bowl. Stir in water, milk and 1 1/2 teaspoons oil until a ball of dough forms. Knead dough on a lightly floured surface 3 minutes. Cover and let rest 15 minutes.
DIVIDE sopaipilla dough into four portions; roll each into a 6 to 7-inch circle.
HEAT 1/4 cup oil to 375ºF degrees in a deep fryer or deep skillet. Fry each circle, one at a time, 2 to 3 minutes on each side or until golden brown. Drain on paper towels.
COMBINE chicken, chili powder, cinnamon, oregano, thyme, cloves, salt and vinegar in medium bowl; mix well. Set aside.
BOIL diced potato in lightly salted water until tender; drain, set aside. Heat 2 tablespoons oil in medium skillet over medium-high heat. Add ground chicken mixture; sauté, stirring frequently to break up pieces, until almost cooked through.
ADD boiled diced potato, diced onion and remaining 1 tablespoon oil; continue to sauté until potatoes begin to brown.
CUT a slit on one side of each sopaipilla; fill with 1/4 of the chicken mixture. Sprinkle tops with additional cheese. Serve with prepared salsa verde.