Preheat the oven to 375 degrees F.
Season the chicken with the salt and pepper.
In a large oven-safe skillet, melt the butter on the stovetop over medium heat.
Add the chicken to the skillet, skin-side-down.
Cook until golden brown, about 5-7 minutes.
Flip and cook until cooked through at 165 degrees F, about 4-5 minutes.
Place the skillet in the oven. Cook for 15-20 minutes, until the chicken is cooked through.
Whenever the chicken is done, transfer it to a plate until ready to serve.
Heat a large pot or Dutch oven over medium heat on the stovetop.
Heat the oil until it shimmers.
Add the garlic and ginger, and cook until fragrant, about 1 minute.
Add in the soy sauce and mirin, and stir.
Add the chicken stock.
Place the lid on the pot until the contents come to a boil.
Remove the lid and simmer, uncovered, for about 5 minutes.
Add in the mushrooms and simmer for 10 minutes to draw some of the mushroom flavors into the broth. Season with salt to taste.
Fill a small pot with enough water to cover the eggs.
Bring the water to a boil.
Lower the eggs into the boiling water.
Simmer until the eggs reach desired doneness, about 7-8 minutes.
Transfer the eggs to a bowl of ice water.
Once cooled, peel and slice in half lengthwise.
Cook the ramen according to package instructions, discarding any flavor packets.
Drain the noodles.
Slice the cooked chicken.
Divide the cooked noodles into two large bowls.
Pour the broth over the noodles.
Top the noodles with the eggs, scallions, and fresh bean sprouts, and a slice of lime.
Serve.