Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and add chicken and 2 cups broth. Close and lock the lid and seal the vent. Choose manual high pressure and set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions for 10 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.
Remove chicken, shred, and set aside. Pour broth into a bowl and reserve.
Turn the empty Instant Pot(R) to Saute and melt butter. Add onion to melted butter and saute 2 to 3 minutes. Add shredded chicken, reserved broth, and remaining broth. Stir in garlic powder, Italian seasoning, salt, and pepper. Place egg noodles on top and press down to submerge; if necessary, add more water just to cover the noodles.
Close and lock the lid and seal the vent. Choose manual high pressure and set timer for 4 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions for 5 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.
Add frozen vegetables, cream of chicken soup, and sour cream to the pot. Stir to combine and replace the lid. Wait 5 minutes for vegetables to warm through.
