Rinse the roasted Hatch chiles with water to remove any excess skin. If desired, remove the stem and seeds from the chiles. Slice the chiles into thick strips, place into an oven-safe bowl. Gently heat the chiles in a warm oven.
Remove the green onion roots and discard. Remove the white end from the green onion and set aside. Finely chop the green onion and set aside.
In a saucepan combine the white onion ends, Miso broth, vinegar, sesame oil and soy sauce. Bring to a boil and reduce heat to a simmer. Cover and cook for 15 minutes. Salt to taste.
Meanwhile, fill a large saucepan with water and bring to a boil. Once boiling, gently place 2 eggs into the water. Cook for 7 to 8 minutes and immediately place the eggs into an ice bath. Peel and slice the eggs just before service.
With 3 minutes remaining in the egg cook time, add the ramen noodles to the boiling water. Cook the ramen (alongside the eggs) for 3 minutes or until desired doneness.
Add the cooked noodles into two shallow bowls and cover with the miso broth. Add the basil leaves, soft boiled eggs and 2 to 3 Hatch chiles to each bowl. Top with green onions and serve warm.