Step 1: Prepare the Chicken Broth

1. Preheat oven to 375°F (191°C).
2. Pat the chicken dry. Mix salt and pepper, then rub the mixture all over the chicken, including under the skin and inside the cavity.
3. Place the chicken in a Dutch oven. Brush with melted butter. Arrange onion, celery, carrots, and garlic around the chicken. Cover with the lid.
4. Roast for 1 hour, then remove the lid and roast for another 30 minutes until golden brown.
5. Transfer the Dutch oven to the stovetop. Add water or broth, fresh herbs (tied as a bouquet garni), and peppercorns. Bring to a boil, then reduce to a simmer. Cover and cook for 3 to 4 hours until the chicken is tender.

Step 2: Make the Egg Noodles

1. In a bowl, mix flour and salt. Make a well in the center and add eggs and yolks. Mix until a crumbly dough forms.
2. Knead on a floured surface until smooth and tacky. Wrap in plastic wrap and let rest for 30 minutes.
3. Divide dough into two portions. Roll each through a pasta machine or with a rolling pin until desired thickness. Cut into thick strands and let dry for at least 10 minutes.

Step 3: Assemble the Soup

1. Remove the chicken, vegetables, and herbs from the broth. Shred the chicken, discarding bones and skin. Strain the broth.
2. In the Dutch oven, melt butter over medium-high heat. Sauté garlic and onion until softened.
3. Add diced celery and carrots, seasoning with salt and pepper. Cook until tender.
4. Return shredded chicken to the pot, stir, then pour in the strained broth. Add water if needed. Bring to a boil, then simmer for 25 to 30 minutes.
5. Add the egg noodles and cook for 7 minutes until tender. Remove from heat and serve hot.