Cook the chicken*: Preheat the oven to 375 degrees. Season the chicken generously with salt and pepper. Melt the butter in a large oven-safe skillet over medium heat. add chicken, skin-side down, and cook until the skin is golden brown and releases easily from the pan, about 5 to 7 minutes. Flip the chicken over and cook for another 4-5 minutes, until golden. Transfer the skillet to the oven and roast for 15-20 minutes, until the chicken is cooked through. Remove from the oven, transfer the chicken to a plate and cover with foil until ready to serve.
Make the ramen broth: Heat the butter in a large pot over medium heat, until shimmering. Add the a small piece of the cooked chicken, garlic and ginger and cook for a few minutes until softened. Add the soy sauce and mirin, and still to combine. Cook for another minute. Add the stock, and bring to slow boil. and let simmer uncovered for 5 minutes. In a new pan add 1 tablespoon butter then quickly saute (three minutes) and add the mushrooms. Simmer gently for another 5 to 7 minutes and season with salt to taste.
Make the soft-boiled eggs: Fill a pot with enough water to cover the eggs, and bring to a boil. Gently lower the eggs into the boiling water, and let simmer for 7 minutes (for a slightly runny yoke) or 8 minutes (for a soft, but set-up yoke).
Meanwhile fill a large bowl with ice water. When the timer finishes, transfer the eggs to the ice bath to stop the cooking process. Wait at least 5 minutes or until cool enough to handle, then carefully peel away the shell and slice in half lengthwise. Set aside until ready to serve.
Assemble the ramen bowls: Meanwhile, chop the scallions and jalapeño (if using). Slide the chicken into thin pieces. Set aside. When the eggs finish cooking, add the ramen noodles to the boiling water. Cook for 2-3 minutes, until soft, then divide the noodles into two large bowls. Add the sliced chicken,ginger,scallions and the ramen broth.