Season the chicken lightly with pepper. Press Sauté on the Instant Pot and add the oil. When hot, sear the chicken on both sides until browned, about 3 minutes per side. Press Cancel.
Add the chicken broth, water, soy sauce, garlic, and the white parts of the green onions. Cover the Instant Pot, select Manual, and set to High pressure for 10 minutes. Quick release the pressure at the end of the cooking time.
Immediately transfer the chicken to a cutting board, removing and discarding skin and bones if using bone-in chicken.. Discard the green onion. Shred the chicken and return it to the pot.
Press Sauté on the Instant Pot and add the mushrooms, submerging them under the liquid. Bring to a boil and simmer about 2 minutes. Add the noodles and cook until soft, another 2 minutes. Press Cancel. Add the bok choy and shredded carrot, stirring to combine. Taste and add seasoning as desired.
Divide the broth into bowls and top with the eggs, the thinly sliced green parts of the green onions, crushed seaweed snack flakes, sesame seeds, and chili oil or sriracha (if desired.) Serve immediately.