Place chicken, carrots, celery, onion, broth, thyme, pepper and 3/4 teaspoon salt in a 6-quart slow cooker. Cover and cook on low for 5 to 6 hours until chicken reaches 165°F and pulls apart easily.
Transfer chicken to a clean plate and cover. Turn the cooker to high and replace the lid for 15 minutes while making the dough.
Mix 1 1/2 cups flour and remaining salt in a bowl. Rub in cold butter until crumbly, then stir in milk, chives and only enough of the 2 tablespoons dough water to form a firm, rollable dough.
Dust the work surface with the 2 tablespoons rolling flour. Roll dough about 1/8 inch thick and cut into strips roughly 1 by 2 inches.
Slip dough strips one at a time into the hot broth, pressing them gently below the surface. Cover and cook on high for 40 to 50 minutes without repeatedly lifting the lid, until a cut strip has no raw flour center.
Shred chicken into large pieces. Mix cornstarch with the 2 tablespoons slurry water and stir into the pot with chicken and evaporated milk.
Cover and cook on high for 10 to 15 minutes until thickened, bubbling at the edges and chicken is 165°F. Divide among 6 bowls.