Preheat the oven to 375 degrees. In a large bowl, combine the ground beef, breadcrumbs, egg, milk, salt, and pepper until evenly mixed. Set aside.
Heat 1 tablespoon of olive oil in a skillet over medium heat. Add the sliced mushrooms and cook until they release their liquid and turn golden, about 6 minutes. Remove from heat.
On a sheet of parchment paper, press the meat mixture into a rectangle about 10 by 8 inches. Lay the Swiss cheese slices down the center, then top with the cooked mushrooms. Fold the sides of the meat over the filling and shape into a loaf, sealing the edges.
Place the loaf on a baking sheet. Toss the carrots, parsnips, and onion with the remaining olive oil and arrange them around the meatloaf. Bake for 50 to 60 minutes until the meatloaf reaches 160 degrees internally and the vegetables are tender.
Let the meatloaf rest for 10 minutes before slicing and serving with the roasted vegetables.