Preheat oven to 400° with a rack in top third of oven. Put pumpkin on a greased rimmed baking sheet. Roast pumpkin, covered, until tender when pierced with a fork, 20 to 25 minutes.

Meanwhile, heat oil in a large, wide pot over high heat. Add lamb or beef, 1 tsp. salt, and the pepper. Cook, stirring as needed, until meat is browned on all sides. Add onion and garlic and cook until vegetables are starting to soften, about 3 minutes. Transfer lamb or beef  mixture with juices to a bowl; add carrots and mushrooms.

Reduce heat to medium-low, whisking constantly, until it reduces, about 1 minute. Pour in wine and broth and simmer, stirring frequently, until thickened, about 4 minutes. Pour over lamb mixture, stirring to combine. Spoon mixture into 6 individual ramekins.

Mash pumpkin in a bowl with the grated parmesan until mixture is consistency of mashed potatoes. Dollop over lamb or beef .

Bake until browned and sauce is bubbling, about 30 minutes. Let sit 10 minutes; sprinkle with parsley.