Heat the oil in a large skillet over medium heat. Add the diced onion and bell peppers and cook, stirring occasionally, until softened and lightly golden, about 5 to 7 minutes. Add the minced garlic and stir for 30 to 60 seconds until fragrant.

Stir in the tomato paste, cumin, oregano, salt, and pepper. Cook, stirring, for about 2 minutes until the tomato paste darkens and the spices smell toasty. Do not rush this step as it builds the deep flavor base of the filling.

Add the shredded chicken and water. Stir to combine and simmer until most of the liquid evaporates and the chicken is juicy and well-coated, about 5 to 8 minutes. Remove from heat and let cool completely. Chill in the fridge for at least 30 minutes for best results.

Preheat oven to 400 degrees F. Line a baking sheet with parchment paper. Place thawed dough rounds on a clean work surface and spoon about 2 tablespoons of filling onto the center of each round. Do not overfill.

Moisten the edges of each dough round with water. Fold one half over the filling to form a half-moon shape. Press edges firmly together then crimp all the way around with a fork to fully seal. A good seal prevents leaks during baking.

Brush the tops with the beaten egg. Bake for 25 to 30 minutes, until deep golden brown and crispy. Let cool a few minutes before serving as the filling holds a lot of heat inside.