Carefully slice the elk into 8 steaks, roughly ½ inch thick. It’s easier to slice when the meat is partially frozen. Place each slice between 2 pieces of plastic wrap and pound with a meat mallet to about ¼ inch thickness. Season both sides with kosher salt and freshly ground black pepper.
Lightly coat each steak in flour, shaking off any excess. Set aside on a plate.
In a large skillet, heat 4 tablespoons butter over medium heat. Working in batches, cook the elk until golden brown, about 2 minutes per side. Remove the steaks and set aside.
Add the remaining butter and shallots to the skillet. Cook until softened, about 2–3 minutes. Add white wine, scraping up any browned bits from the skillet. Stir in capers and the juice from ½ a lemon, cooking until the sauce reduces by half. Taste and adjust seasoning with salt and pepper if needed.
Transfer the elk to a serving platter, pour the sauce over the top, and garnish with sliced lemons. Serve with cooked thin spaghetti or your favorite side.