In a skillet over medium heat, add the neutral oil. Once hot, toss in the diced yellow onion and bell peppers. Sauté for about 5-7 minutes until the onion becomes translucent and the peppers soften.
Next, add the minced garlic, tomato paste, ground cumin, dried oregano, kosher salt, and ground black pepper to the skillet. Stir to combine and cook for an additional 2-3 minutes, letting the spices bloom and infuse the vegetables with flavor.
Add the shredded cooked chicken to the skillet, followed by the water. Stir everything together and let it simmer for about 5 minutes, allowing the flavors to meld. Once done, remove from heat and let the mixture cool slightly.
Preheat your oven to 400°F (200°C). On a clean surface, lay out the empanada dough rounds. Spoon a generous amount of the chicken filling onto one half of each dough round, leaving the edges clear for sealing.
Fold the dough over to create a half-moon shape. Press the edges together firmly to seal, and use a fork to crimp the edges for an extra decorative touch.
Place the empanadas on a baking sheet lined with parchment paper. Brush the tops with the lightly beaten egg for a beautiful golden finish. Bake in the preheated oven for 20-25 minutes, or until they are golden brown and crispy.