Heat 2 tablespoons of vegetable oil in a large saute pan over medium-high heat. Add onions and saute until slightly softened, about 2 minutes. Add the garlic, chili powder, oregano, and cumin and cook, stirring, until fragrant, about 1 minute. Add the shredded chicken. Season with salt and pepper. Stir in the green chiles. Cook mixture for about 5 minutes, or until heated through. Set aside and cool slightly.
One at a time, place the tortillas on a work surface, keeping the remaining tortillas covered with a damp towel. (For pliable tortillas, sprinkle lightly with 1/4 teaspoon water and cover with a damp paper towel. Microwave for 10 to 20 seconds just before ready to stuff, and keep covered until ready to roll.) Spread 1/2 cup of the chicken mixture in a rectangle in the center of the tortilla. Sprinkle with 1 to 2 tablespoons of cheese. Fold the bottom of the tortilla over the filling and then fold the sides over. Fold the top of the tortilla over the folded sides and secure the sides with wooden toothpicks. Place folded side down.
Heat the remaining vegetable oil in a deep stockpot or deep fryer to 360 degrees F. Using tongs, lower the chimichangas folded side down, 1 at a time into the hot oil. (It is helpful to hold the chimichanga with the tongs until it has "sealed" together.) Cook until golden brown on all sides, about 3 minutes. Drain on a paper towel-lined plate and continue with remaining chimichangas.
Remove toothpicks from the chimichangas and serve immediately with pre-made enchilada sauce or salsa drizzled over top and a sprinkling of cheese.