Place the whole chicken in your largest stockpot. Add the cold water, celery chunks, carrot chunks, quartered onion, smashed garlic, bay leaves, peppercorns, salt, and thyme. Bring to a boil over medium-high heat, then immediately reduce to a low simmer. Skim off any foam that rises in the first 20 minutes. Simmer for about 90 minutes.

Carefully lift the chicken out of the broth using tongs and a slotted spoon. The chicken should have reached 165°F internally at the thickest part of the thigh (verify with meat thermometer). Set on a cutting board to cool slightly.

Shred the chicken off the bones using two forks or your hands. You should get about 2–3 cups of meat. Verify there's no pink and that it was cooked to 165°F internally.

In the same pot (rinsed out), melt the butter over medium heat. Add the diced onion, sliced carrots, and sliced celery. Cook for 5-6 minutes until the onion is soft and translucent. Add the minced garlic and cook for 30 seconds until fragrant.

Pour the strained broth back into the pot. Bring to a simmer.

For homemade noodles: Mound the flour on a clean surface, make a well in the center, and crack in the eggs. Add milk and salt. Beat eggs with a fork, gradually pulling flour from the edges until a shaggy dough forms. Knead for 5 minutes until smooth. Wrap in plastic and rest 20 minutes. Roll out to 1/8 inch thick, roll up loosely, and cut into 1/4-inch strips. For store-bought: follow package directions.

Add noodles to the simmering broth. Fresh noodles cook in 3-4 minutes (they float when done). Store-bought take 6-8 minutes. Add the shredded chicken and chopped parsley. Taste and adjust salt, add pepper, and squeeze in a little lemon juice if using.

Ladle into big bowls and serve with buttered bread, biscuits, or crackers.