Place the chicken in the Instant Pot and pour in water so the chicken is totally covered. Add the onion, garlic, bouillon, Italian seasoning, lemon pepper seasoning, bay leaves, and a pinch of salt and pepper. Stir, then secure the lid and move the valve to the sealing position. Hit Manual or Pressure Cook on High Pressure for 10 minutes. Quick Release when done.
While the pot’s cooking, cook the egg noodles according to package directions and set aside.
When the lid comes off the pot, remove the chicken with tongs and set aside to cool. Remove and discard the bay leaves and onion (if you wish) using a slotted spoon.
Add all remaining ingredients to the Instant Pot except for the chicken and egg noodles, and stir. Secure the lid and move the valve to the Sealing position. Hit Keep Warm/Cancel followed by Manual or Pressure Cook on High Pressure for 5 minutes. Quick Release when done.
While the pot is cooking the soup, pick the slightly cooled chicken meat from the bones and discard the bones, skin, and cartilage. Shred the meat by hand and set aside.
Stir in the shredded chicken and serve in bowls topped with egg noodles.