In a medium saucepan, combine the stock, miso and soy sauce; bring to a boil over high heat, whisking to dissolve the miso. Add the shredded chicken and simmer over moderate heat for 2 minutes. Keep hot over low heat.

Set up a small ice bath. Fill another medium saucepan with water and bring to a boil. Add the eggs and simmer over moderate heat for exactly 7 minutes. Using a slotted spoon, transfer the eggs to the ice bath to cool. Carefully peel the eggs and cut them in half lengthwise.

Meanwhile, return the saucepan of water to a boil. Add the ramen and cook until just softened, about 3 minutes. Drain well and transfer to 2 large bowls. Ladle the broth and chicken over the noodles and top with the eggs. Serve with thinly sliced chicken and green chilies.