Set the Instant pot on “Saute” mode. Season both sides of the chicken thighs with salt and pepper. Add 1 tablespoon of oil to the bottom of the pot. Brown the chicken thighs for 2-3 minutes on each side; you may need to do this in 2 batches. Remove the chicken thighs from the pot and set aside on a plate.
Add the jalapeno and minced garlic to the pot. Stir until fragrant, about 30 seconds. Add the salsa verde, chile pepper, oregano, cumin, and hominy, stirring to coat the hominy. Add the chicken stock and return the chicken thighs to the pot.
Place the lid on the Instant Pot and set the vent to “sealing.” Cook on manual high pressure for 17 minutes. Allow for a 10-minute natural pressure release, and then manually release the rest of the pressure.
For the slow cooker:
Follow the steps above for sauteeing all of the ingredients, except do this in the bottom of a skillet or heavy-bottomed pot. Next, add all ingredients (except the hominy) to the slow cooker. Cook on low for 6-8 hours or on high for 3-4 hours until the chicken is cooked through and easily shredded with a fork. Add the hominy during the last hour of cooking. Proceed with the recipe as written.
Finishing the soup:
Remove the chicken thighs from the pot and shred with two forks. Transfer the shredded chicken back into the pot.
Serve the soup garnished as desired with fresh lime juice, a sprinkling of Cotija cheese, sliced avocado, fresh cilantro, and crushed tortilla chips.
Leftover soup can be refrigerated for up to 5 days and reheated as needed, or frozen for up to 6 months.