Mince the garlic, grate the ginger, slice the mushrooms 1/4 inch thick, cut the carrots into thin matchsticks, halve and rinse the bok choy, and slice the green onions. Set the ramen noodle blocks aside and discard the seasoning packets. Fill a large bowl with ice and cold water.
Bring a medium saucepan of water to a rolling boil. Lower the eggs in with a spider strainer and cook for 6 1/2 minutes for jammy yolks (7 minutes for firmer). Transfer to the ice bath for 5 minutes, then peel under cool running water.
Heat the vegetable oil in a large Dutch oven over medium-high heat. Add the mushrooms in a single layer and cook 5 to 6 minutes until golden brown. Reduce heat to medium, add the garlic and ginger, and stir constantly for 30 to 45 seconds until fragrant.
Pour in the chicken broth and water, scraping up browned bits from the bottom. Stir in the soy sauce and mirin, then bring to a gentle boil over high heat.
Lower the chicken breasts into the broth, return to a simmer, then reduce heat to low, cover, and cook 14 to 18 minutes until the thickest part reaches 165°F. Rest the chicken 5 minutes, then shred with two forks. Skim any foam from the broth.
Add the carrot matchsticks to the simmering broth and cook about 4 minutes until crisp-tender.
Add the ramen noodle blocks, pressing them down to submerge. Simmer 2 minutes, loosening with tongs, then nestle in the bok choy and cook 1 to 2 minutes more. Remove the pot from the heat.
Return the shredded chicken to the pot and stir about 1 minute to warm through. Stir in the toasted sesame oil, then taste and season with kosher salt and black pepper as needed.
Divide the noodles, carrots, and bok choy among 4 bowls. Ladle the hot broth and chicken over the top, tuck in two soft-boiled egg halves per bowl, and sprinkle with green onions and sesame seeds. Add chili crisp if desired and serve right away.