Turn on a multi-functional pressure cooker (such as Instant Pot®), add oil, and select Saute function. Add chicken, pesto, garlic powder, salt, and pepper to the hot pot, then pour in chicken broth.
Place onion, bell pepper, and carrots on top of chicken and season with salt and pepper. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.
Meanwhile, melt butter in a skillet over medium heat and cook until golden. Stir in flour until mixture thickens. Add garlic, oregano, and sage while continuously stirring, as a roux burns fast. Slowly pour in 2 cups of milk, stirring until all clumps are gone and sauce it thick, adding more milk as needed.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes.
Remove chicken from the Instant Pot®, let cool for 5 minutes, and dice into small cubes.
While you are waiting for the chicken to cool, add gnocchi, basil, and roux mixture to the Instant Pot®. Close and lock the lid. Select low pressure according to manufacturer's instructions; set timer for 1 minute. Allow 10 to 15 minutes for pressure to build.
Slice spinach into strips; add to the cooker with chicken. Mix for 2 minutes to heat through before serving; the heat from the soup will cook the spinach quickly.