Melt butter in a large, deep skillet over medium heat. Add carrots, celery, onion, and 1/2 teaspoon salt. Cook until vegetables soften, 6-8 minutes.
Add mushrooms and garlic, cook until mushrooms release their moisture, about 5 minutes.
Sprinkle in flour, stir to coat vegetables. Cook 2 minutes, stirring constantly.
Gradually whisk in stock, then milk, stirring continuously to prevent lumps. Add Dijon, Italian seasoning, and nutmeg if using.
Bring to gentle simmer and add gnocchi. Cook uncovered for 4-5 minutes, stirring occasionally.
Stir in chicken and peas. Simmer until heated through and sauce thickens, about 3-4 minutes.
Fold in fresh herbs. Season to taste with salt and pepper.
Let rest 5 minutes before serving. Top with Parmesan if desired.