Heat a splash of vegetable oil in a skillet over medium heat. Add diced onions and minced garlic, sauté until translucent. Stir in chili powder, oregano, ground cumin, salt, pepper, and diced green chilies. Fold in shredded chicken until well mixed.
Take a warmed tortilla, place about half a cup of the chicken mixture in the center, sprinkle with Monterey Jack cheese, fold sides inward, and roll tightly. Repeat until all tortillas are filled.
For frying, heat vegetable oil in a deep skillet over medium-high heat. Fry chimichangas seam-side down until golden brown, about 3-4 minutes per side. For baking, preheat oven to 425°F, place chimichangas seam-side down on a baking sheet, spray with cooking oil, and bake for about 20 minutes until crispy.
Serve warm with salsa, sour cream, guacamole, pico de gallo, or any toppings you desire.