Melt butter in a Dutch oven or large pot over medium heat.
Add onion, carrots, celery, and mushrooms. Cook until soft and mushrooms release liquid.
Season with salt, pepper, poultry seasoning, and thyme. Stir well.
Add garlic and cook for 1 minute until fragrant.
Sprinkle flour over vegetables and stir. Cook for 1 minute to remove raw taste.
Slowly pour in chicken broth while stirring until smooth.
Add milk or half-and-half and simmer until the sauce thickens.
Stir in gnocchi and simmer for 5–6 minutes until tender.
Add cooked chicken and frozen peas. Stir and heat through.
Remove from heat and let rest for a few minutes before serving.