Mix the chicken, gochujang, lime juice and season with salt and pepper. Set aside.
Heat a skillet or saute pan over medium heat. Once hot, add half of the oil. Saute the chicken until nicely brown all over then remove it from the pan and set aside. Add the remaining oil to the pan. Saute the onion, garlic, and ginger until they start to soften.
Stir in the broth and dashi, bring to a simmer. Scrape the bottom of the pan to release any bits of flavor. Add in the eggs and cover with a lid. Simmer for 5 minutes (for a slightly-runny yoke) or 8 minutes (for a hard yoke). Remove with a slotted spoon and place in a bowl of room temp water. Taste and adjust the seasoning with salt if needed.
Add the noodles into the pan, cover with a lid and cook according to the directions on the package. Toss in the chicken to warm.
Peel the eggs and cut in half.
Portion the noodles (if there is any broth remaining spoon 2 tablespoons over top and discard the rest). Top with chicken and eggs. Garnish with snow peas and sesame seeds.