Place chicken in the bottom of a large Dutch Oven or soup pot then add chicken broth, carrot, celery, shallots or onion, dried thyme, dried parsley, bay leaves, garlic, salt, and pepper. Bring to a boil then place a lid on top, lower heat and simmer for 30 minutes or until chicken is tender. Remove chicken from pot then shred when cool enough to handle and set aside.
Meanwhile, make the noodles. Whisk together eggs, extra egg yolk, salt, and water in a small bowl. Add flour to a large bowl then make a well in the center. Pour egg mixture into the well then using a fork, whisk around the edges to gradually add flour to the egg mixture from the sides until a stiff dough has formed. Place dough onto a heavily floured surface then knead until dough comes together and is smooth.
Cut dough ball in half then roll each half until very thin, sprinkling with flour and flipping/rotating the dough sheet occasionally to avoid sticking. Using a pizza cutter, cut dough into 2-1/2" by 1/2" strips then toss with extra flour to avoid sticking and place in a container until ready to use.
Once chicken has been removed from the pot, sprinkle in the noodles then cook until plump and tender, 13-15 minutes, stirring often to avoid sticking to the bottom of the pot. If liquid gets too thick, add a splash of extra chicken broth.
Add frozen peas and shredded chicken to the pot. To a jar with a tight fitting lid add milk and flour then shake vigorously to combine. Stream into Dutch oven then cook until mixture is thick and bubbly, 5 minutes. Taste then add salt and pepper then let chicken and noodles sit off the heat for 10 minutes to thicken before serving.
Notes
I like Bob's Red Mill Gluten Free 1-to-1 Baking Flour for this recipe though I have made the noodles with many different types of gluten free flour blends with success. Whatever blend you use, make sure it includes a binder such as xanthan gum.