In a large Dutch oven or soup pot over medium-high heat, add the chicken pieces, 6 cups of the broth, the carrots, the celery, the shallots, the dried thyme, the dried parsley, the bay leaves, the garlic, the extra salt, and the pepper and bring the mixture to a boil.
Cover the pot and lower the heat to a simmer.
Simmer the soup mixture until the chicken pieces are tender and reach an internal temperature of 165 degrees F, about 30 minutes.
Transfer the chicken pieces from the pot and allow them to cool until safe to handle.
Remove the bones and shred the chicken using two forks.
In a small bowl, add the eggs, the egg yolk, 1 teaspoon of the salt, and the water and whisk to combine.
In a large bowl, add 2 cups of the flour and make a well in the center.
Pour the egg mixture into the well in the flour and using a fork, whisk around the edges of the well to gradually add the flour to the egg mixture until incorporated and a stiff dough forms.
Using the extra flour, heavily flour a work surface.
Place the dough onto the prepared work surface and knead it until it comes together and is smooth.
Form the dough into a ball and then cut the ball in half.
Roll each of the dough halves until very thin, sprinkling them with the extra flour and rotating the dough to avoid sticking.
Cut the dough halves into 2 1/2x1/2-inch strips.
Toss the dough strips with the extra flour to avoid sticking.
Sprinkle the noodles into the pot with the soup and cook, while stirring often, until they are plump and tender, about 13-15 minutes. If the soup begins to thicken too much, add the extra broth, as needed.
Add the frozen peas and the shredded chicken to the pot and stir to combine.
In a small bowl, add the milk and the remaining 2 tablespoons of the flour and whisk vigorously to combine.
Stream the slurry mixture into the pot and cook until the soup mixture is thick and bubbly, about 5 minutes.
Taste the soup for seasoning, adding more of the salt and the pepper, as needed.
Transfer the pot from the heat and let it thicken for 10 minutes.
Serve.