Place chicken breasts, water, and onion in a multi-functional pressure cooker (such as Instant Pot®). Close and lock the lid. Select high pressure according to manufacturer’s instructions; set timer for 30 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer’s instructions, about 5 minutes. Unlock and remove the lid. Drain off water. Shred chicken using 2 forks. Mix in green chiles.
Combine chicken mixture with Monterey Jack cheese. Spoon some of the chicken filling into the middle of each tortilla and roll up tightly. Grease a 9×13-inch casserole dish; arrange filled tortillas in the dish.
Preheat the oven to 350 degrees F (175 degrees C).
Melt butter in a saucepan over medium heat. Stir in flour and cook for 1 minute. Add water, chicken stock concentrate, and cumin; whisk until smooth. Cook over medium heat until thick and bubbly, about 5 minutes. Stir in Greek yogurt, pepper Jack cheese, and Monterey Jack cheese. Do not boil, as yogurt may curdle.
Pour sauce over enchiladas in the casserole dish.
Bake in the preheated oven until cheeses melt, about 22 minutes. Broil on high until top browns, about 3 minutes.
If freezing, do not make sauce. You’ll want to freeze the casserole dish and, once frozen, transfer the enchiladas to a large resealable plastic bag. Make the sauce right before you plan to bake the enchiladas.
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