Heat butter and olive oil in a large Dutch oven over medium heat. Add onion, carrots, and celery. Cook 6 to 8 minutes until onion is translucent and carrots have softened slightly. Add minced garlic and stir 1 minute until fragrant.
Sprinkle flour over the vegetables. Stir continuously for 1 to 2 minutes until fully absorbed, no dry patches remain, and the mixture smells slightly nutty.
Pour in chicken broth 1 cup at a time, stirring after each addition until smooth. Add whole milk and stir to combine. Increase heat to medium-high and bring to a gentle boil, stirring occasionally, until the broth thickens visibly, about 5 to 7 minutes.
⚠️ Food Safety: If adding raw chicken here, confirm internal temperature reaches 75°C (165°F) before continuing. Add shredded chicken, dried thyme, dried parsley, bay leaves, ½ teaspoon salt, and ¼ teaspoon black pepper. Simmer on medium for 10 minutes.
Add egg noodles. Cook 8 to 10 minutes, stirring every 2 to 3 minutes, until noodles are tender with a slight bite.
Reduce heat to low. Remove and discard bay leaves. Pour in heavy cream and stir gently for 1 to 2 minutes. Simmer on low for 3 to 4 minutes until fully incorporated. Do not boil.
Taste and adjust salt and pepper. Ladle into bowls, garnish with fresh parsley, and serve immediately.