In a large skillet over medium-high heat, melt butter. Add mushrooms, onion, carrot, and celery and cook, stirring often, until softened, 8 to 10 minutes.
Add flour and quickly stir to coat vegetables. Immediately whisk in broth, milk, 1 ½ tsp. salt, and 1 tsp. pepper. Bring to a boil over high heat, then reduce heat to medium and simmer, stirring often, until thickened, 3 to 5 minutes. Fold in turkey and peas and cook, stirring occasionally, until warmed through, 4 to 6 minutes.
Divide turkey mixture among plates. Top with parsley. Serve with biscuits alongside.
