In a large pot, combine all of the ingredients for the broth. Bring to a boil, turn temperature down to low and simmer for about 30 minutes, covered, while the rest of the ingredients are prepared. Be sure to really gently simmer so that you don't reduce the broth by too much.
Bring a small pot of water to a boil. Turn the heat to medium (so that the boil is not too rapid) and add the eggs. Cook until soft-boiled, typically 6.5 minutes. If you live in altitude like I do, it will be closer to 8 minutes. Remove and set into an ice bath for at least 15 minutes.
Using a sharp knife, slice the chicken breasts in half, creating two thin cutlets. Using a meat mallet (or a glass), pound the chicken cutlets until they are about 1/2 inch thick.
Set up a dredging station by pouring the panko on a plate, seasoning with a pinch of salt and pepper, and whisking 1 egg in a small bowl.
Dredge the chicken cutlets by dipping in the egg, then covering in panko, being sure to shake off any excess before placing on a plate.
Heat a skillet over medium heat for a few minutes. Add 1-2 tbsp of oil and continue to heat.
Add the chicken cutlets to the skillet, cooking for 3-4 minutes per side until golden brown on each side. Set aside on a plate. Rest for at least 10 minutes and then slice thinly in diagonal slices.
In the same or different skillet heated to medium and with another drizzle of oil, add the shiitake mushrooms and sauté to brown. When almost done cooking, add 1 tbsp coconut aminos or low-sodium soy sauce to glaze. Set aside.
Cook ramen noodles according to package.
Assemble ramen by placing noodles in a bowl, spooning in the hot broth, and adding the toppings. Top with green onion, sesame seeds and lime for garnish.