Set Instant Pot to Sauté on Normal mode. Drizzle 1 tablespoon olive oil across bottom surface of Instant Pot insert, then heat oil until shimmery.
When oil is hot, add 1 medium white onion and 3 cloves garlic. Stir to incorporate, then sauté, stirring occasionally, approximately 5 minutes or until onions have softened.
Add 1 teaspoon cumin, 2 teaspoons chili powder, and 1 teaspoon dried oregano and stir until well incorporated.
Pour in 1 28-ounce can crushed tomatoes and 1 14-ounce can low-sodium chicken stock. Stir to incorporate, making sure to scrape bottom of insert to loosen any garlic or onions that may have stuck to pot.
Add 2 large boneless, skinless chicken breast cutlets to Instant Pot and gently stir to incorporate.
Secure lid on Instant Pot with vent in Sealing position. Set Instant Pot to Manual High Pressure and set cook time to 18 minutes. Instant Pot will pressurize, then cook time will begin.
Once 18-minute cook time has ended, Natural Release pressure 10 minutes. After 10 minutes, carefully Quick Release any remaining pressure, then remove lid and set aside. Use caution - any escaping steam will be very hot.
Set Instant Pot to Sauté on Less mode. Transfer chicken from Instant Pot to cutting board and set aside.
Add 1 15-ounce can black beans, 1 cup whole corn kernels, and 2 4-ounce cans chopped green chiles to Instant Pot. Stir to incorporate, then simmer mixture 10 minutes.
While mixture simmers, use 2 forks to shred chicken into pieces of preferred size. See Notes for alternate shredding methods.
After 10 minutes, turn off Instant Pot and let cool slightly. Add ½ cup full-fat sour cream to soup and stir until sour cream is fully incorporated.
Return shredded chicken to pot, then taste soup and add salt or adjust cumin, chili powder, or oregano as desired.
Ladle soup into serving bowls. Top each serving with Monterey Jack cheese, avocado slices, crushed tortilla chips, chopped cilantro, and/or chopped green onions if desired. Serve warm.