Preheat oven to 350 degrees F. Spray a 9-by-13-inch baking dish with nonstick cooking spray.
Melt the butter in a large skillet over medium heat. Add the onion, sage, and thyme and saute until tender, 4 to 5 minutes. Stir in the flour and cook out the raw flour flavor, an additional 3 minutes. Add the peas and carrots, chicken broth and Dijon mustard and season with salt and pepper. Bring the mixture to a simmer and allow to begin thickening, 5 to 7 minutes. Stir in the pork or chicken and transfer to prepared baking dish. Top with the mashed potatoes (if using store-bought frozen, follow the instructions included on the bag) and spread evenly over the filling. Sprinkle with Parmesan and sprinkle with more pepper.
Bake until the potatoes are lightly golden brown and the filling is hot and bubbly, 50 to 60 minutes.