Preheat the oven to 400°F.
Heat a large skillet over medium heat and add the butter and olive oil. Once the butter has melted, add the carrots, celery, onion, potatoes, and garlic. Season with granulated garlic powder, tarragon if using, and kosher salt. Cook until the vegetables are softened, about 10 minutes. Remove from heat and drain any excess oil if needed.
Add the ground venison to the skillet and cook until browned. Remove from heat, then stir in the cooked vegetables until well combined.
On a lightly floured surface, roll out the puff pastry to roughly 16" x 16". Spread 1 cup of shredded cheese down the center, then layer the ground meat and vegetable mixture on top. Sprinkle with the remaining shredded cheese.
Fold one side of the pastry over the filling, then fold the other side on top to seal. Fold the edges underneath to fully enclose the mixture. Carefully flip the Wellington seam-side down onto a parchment-lined baking sheet.
Brush the top with olive oil and sprinkle with Parmesan, a pinch of kosher salt, and coarse black pepper. Bake for 30-35 minutes or until the pastry is golden brown.
Remove from the oven and let rest for 10 minutes before slicing. Serve warm, with brown gravy if desired.