Heat the oil in a skillet on medium-low and sauté onions until softened and translucent, about 8 minutes. Add in garlic and cook 1 to 2 minutes. Then place the onions and garlic along with the green chiles into a bowl with the cooked and shredded chicken, stir to combine.
Place 4 toasted tortillas into the bottom of a 10x2 square metal pan (an 8x8 pan can be used if in a pinch, just overlap more). Spread half of the chicken/onion mixture on top of the tortillas, then 1/2 cup of salsa, 1/3 cup sour cream, 2 tablespoons cotija and 1/3 of the Monterey Jack cheese. Repeat one more time; 4 more tortillas, the remaining chicken mixture, 1/2 cup salsa, 1/3 cup sour cream, 2 tablespoons cotija and half of the remaining Monterey Jack cheese.
Then, for the last layer, place the remaining tortillas on top and spread the salsa and sour cream, then sprinkle with the last of the cotija and Monterey Jack cheese.
Bake on the middle rack of your oven for 35 to 40 minutes or until the cheese is golden brown on top.