Bring chicken stock to a boil; cook chicken breasts at a boil until no longer pink in the center, 15 to 20 minutes. Remove chicken, cool, and cut meat from bone; keep chicken stock at a low simmer.

Mix flour and 1/4 teaspoon salt together using the dough hook attachment in the bowl of a stand mixer on Low; add eggs and water, 1 tablespoon at time, until dough holds together.

Split dough in half. Roll 1 dough half on a lightly floured surface to 1/8-inch thickness. Cut dough into noodles 2 to 3-inches long x 1/4-inch wide using a sharp knife dipped in flour or a fettuccine pasta cutter. Repeat with other dough half.

Return chicken to the simmering stock; add noodles. Season stock with black pepper and 1 pinch salt. Cook soup until noodles are cooked through, 15 to 25 minutes.