Preheat oven to 375°F. Line a baking sheet with parchment paper.
Heat a large skillet over medium heat. Add olive oil and butter. Once melted, stir in the diced potatoes, carrots, celery, and onion. Cook for 8–10 minutes, stirring occasionally, until the vegetables begin to soften. Add the minced garlic and cook 30 seconds more.
Add the ground venison to the skillet and cook, breaking it apart, until browned and fully cooked.
Season with fennel, granulated garlic powder, kosher salt, and black pepper. Stir well to combine, then remove from heat and allow the filling to cool slightly.
Lightly flour a surface and gently roll out the puff pastry sheet. Sprinkle the shredded cheese evenly over the center of the pastry.
Spoon the venison mixture over the cheese, leaving about a 1-inch border around the edges. Fold the pastry over the filling, seal the edges tightly, and carefully flip seam-side down onto the prepared baking sheet.
Whisk together the egg and water, then brush evenly over the top of the pastry.
Bake for 40–45 minutes, until puffed and deeply golden brown.
Let rest for 5–10 minutes before slicing. Serve warm with rich brown gravy spooned over the top.