Melt the butter in a large sauté pan over medium heat.
Add the diced onion, celery, and carrots. Cook, stirring occasionally, until vegetables are tender, about 8–10 minutes.
Add the garlic and cook for 30 seconds, stirring constantly.
Sprinkle the flour evenly over the vegetables and cook, stirring constantly, for 2 minutes.
Slowly pour in the chicken broth and white wine, stirring and scraping the bottom of the pan to prevent lumps.
Add salt, black pepper, and nutmeg.
Bring the mixture to a boil, then reduce to a simmer and cook until thickened to a gravy-like consistency, about 3–5 minutes.
Remove from heat and stir in heavy cream, thyme, parsley, shredded chicken, and peas.
Taste and adjust seasoning as needed.