Set a programmable pressure multicooker (such as Instant Pot) to Saute. Select high temperature setting, add sesame oil to cooker, and allow to preheat 1 to 2 minutes. Add mushrooms and green onions; cook, stirring occasionally, until browned, about 8 minutes. Press "Cancel."
Add broth and soy sauce, scraping to loosen any browned bits from bottom of pot.
Cover cooker with lid, and lock in place. Turn steam release handle to sealing position. Select manual/pressure-cook setting. Set to cook on high pressure for 10 minutes. (It will take 5 to 10 minutes for cooker to come up to pressure before cooking begins.)
Meanwhile, add water to a saucepan to a depth of 3 in., and bring to a boil over medium-high heat. Add eggs; cook, undisturbed, 6 to 8 minutes (or 10 minutes for hard yolks). Transfer eggs to an ice bath, and let cool 2 minutes.
When pressure cooker is finished, carefully turn steam-release handle to venting position, and let steam escape; float valve will drop. (This will take 3 to 4 minutes.) Remove lid from cooker. Select Saute setting. Select high temperature setting, and bring to a boil. Add noodles, bok choy, and carrots, making sure noodles are submerged; cook, covered with lid resting on top, undisturbed, until noodles are tender, 3 to 4 minutes.
Stir mixture, and divide evenly among four shallow bowls. Carefully peel eggs, and cut in half lengthwise. Top each bowl with an egg; garnish with sesame seeds, if using, and additional green onions and sesame oil.
