Preheat oven to 400 degrees.
Line two baking sheets with parchment or silicone baking mats and set aside.
In a large skillet over medium heat, warm vegetable oil till it begins to shimmer. Add diced onions and cook until translucent, but not brown, about 4 -5 minutes.
Add minced garlic to the pan and cook 30 seconds or until garlic is fragrant.
Add cooked chicken, Hatch Chiles, Hatch chile powder, cumin, oregano, salt and pepper and continue cooking an additional 1-2 minutes until mixture is warmed through.
Lay puff pastry squares out on prepared baking sheets. Mine fit 5 squares to one baking sheet.
Place one heaping tablespoon chicken chile mix in the center of each puff pastry square, being careful not to get too close to the edges. Add 1 teaspoon crumbled Cojita Cheese on top of the chicken.
Lightly brush the edges of the pastry squares with the beaten egg. Fold the square in half over the filling to make a triangle. Crimp the edges of each triangle with a fork to seal. Brush the tops of each pastry triangle with the rest of the beaten egg. Bake in 400 degree oven for approximately 20 minutes or until pastry is golden brown and very puffy.