Mix flour, salt, baking powder, and baking soda together in a large bowl. Add water and 1/4 cup vegetable oil; mix with an electric mixer until well combined and a dough begins to form, about 3 minutes. Remove dough and knead on a lightly floured surface for about 3 minutes. Wrap in plastic wrap and set aside for 15 minutes.

Meanwhile, heat 2 tablespoons oil in a large skillet over medium-high heat. Add shredded chicken and saute until heated through, about 2 minutes. Add onion, bell pepper, cilantro, olives, garlic, sazon seasoning, tomato paste, salt, and pepper and cook, stirring occasionally, until vegetables have softened, about 10 minutes. Remove from heat and let cool.

Divide empanada dough into 12 pieces and form each one into a 4-inch circle. Place about 2 tablespoons filling in the center of each circle, fold in half, and press down on the edges with a fork to seal.

Heat 1 cup oil in a frying pan over medium heat. Add empanadas, working in batches, and fry until golden brown, about 5 minutes per side. Transfer to a paper towel-lined plate.