Begin SAUTÉ mode on your Instant Pot. Drizzle the oil into the pan and let it heat up for a few moments; just until the oil is shimmery. Sprinkle the chicken on both sides with salt and pepper and use tongs to lay them in the hot pan in a single layer.
Brown the chicken on both sides, about 3 minutes per side.
Add in the ginger and garlic and stir for 2 minutes, or until fragrant. Stir in the white part of the green onions, the chicken broth, the soy sauce, and the dried mushrooms, if you're using them.
Lock the Instant Pot's lid in place, switch to MANUAL setting and program in 10 minutes on HIGH. When the cycle is finished, use QUICK RELEASE to vent the steam.
When the pressure valve has dropped back into the lid, carefully remove the lid. Use a slotted spoon to transfer the chicken and green onion whites into a heat-proof bowl.
Discard the white parts of the green onion and shred the chicken roughly with two forks before returning it to the Instant Pot. Taste the soup. If it needs a little boosting, you can add powdered bouillon or a packet of the instant ramen seasoning to taste.
Return the Instant Pot to SAUTÉ mode, and add in any desired vegetables, cooking them as long as the notes specify.
In a separate saucepan, prepare the instant ramen noodles without seasoning packets and drain.
Divide the noodles between serving dishes, ladle the ramen soup over the noodles, add a halved ramen egg to each bowl, and garnish as desired. Serve immediately!