Place the carrots, onions and celery in the bottom of a slow cooker. Place the whole chicken on top of the vegetables and season with salt and pepper. Pour in the white wine and sprinkle basil over the top. Cover and cook on Low setting for 8 to 10 hours.
In a medium bowl, stir together the eggs, water, salt, shortening and flour to form a stiff dough. After I've mixed in as much flour as possible using a fork, I knead the dough with my hand in the bowl to incorporate as much flour as possible. Let the dough rest for a few minutes.
Roll the dough out on a well-floured board to 1/8 inch thickness. Use a pizza cutter or pie crust cutter to cut into strips about 1/2 inch wide and 3 inches long. Dust lightly with flour, and leave to dry for a few hours while the chicken cooks.
When the chicken is done, remove the meat and vegetables to a platter. Transfer the juices to a large pot and stir in 2 quarts of chicken broth. Bring to a boil and add the noodles. Cook for about 10 minutes, until tender. Meanwhile, remove the meat from the chicken and shred. Discard bones and skin. When the noodles are done, return the vegetables to the pot and add shredded chicken meat. Serve. (Better to make a day ahead)