1. Pull the eye-of-rounds from the fridge. Use a sharp knife and slice lengthwise down the center of the muscle to butterfly it open. Be cautious to not cut all the way through. Pat dry and season with salt and pepper.
2. Brush the mushrooms clean using a barely damp paper towel. Do not wash the mushrooms in water. Roughly chop into small, ¼-inch pieces.
3. Heat a large sauté pan over medium-high heat. Add a drizzle of cooking oil and spread the mushrooms and sprigs of thyme across the pan. Once they are down, leave them alone and let them brown. Periodically toss and stir to cook on all sides. About halfway through, or when you think the mushrooms are drying out, add one tablespoon of butter and mix. When the mushrooms are close to being cooked, add the minced garlic. Cook until the garlic is soft and fragrant, which should take a total of 10 to 15 minutes. Remove the mushrooms and set aside on a plate.
4. Spoon the mushrooms inside the eye-of-rounds. Don’t be tempted to overstuff. Save leftover mushrooms for the pan sauce at the end.
5. Using kitchen twine, truss the eye-of-rounds to hold the meat together. Tie a butcher’s knot every 1 to 2 inches across the entire cut of meat.
6. Pour the stock into a small pot over medium heat and bring to a gentle simmer. Reduce the stock for about 10 to 15 minutes. You should end up with about half a cup. This greatly enhances the flavor and can be done in advance.
7. Wipe the sauté pan clean and heat over medium-high heat. Add a tablespoon of cooking oil and lay down the meat once it’s hot. Brown the meat on all sides until it’s a golden color. A small eye-of-round will finish cooking within 6 to 10 minutes, but a larger one might take longer. Reduce the heat as necessary to avoid burning the outside and to cook the meat through to medium-rare. You can’t really use a meat thermometer here because the meat is cut open, but remember that it will cook much faster because it’s cut. If you’re cooking a large eye-of-round, such as an elk’s, you can transfer the meat to a 300-degree oven to finish cooking. Use tongs and squeeze the meat to check how firm or soft it is, the same way you would check a steak for doneness. Once the meat is cooked through, remove to a cutting board and let it rest.
8. Deglaze the pan with cognac. Let the alcohol boil and burn off for a moment, then add the stock. If you have any reserved mushrooms, add them to the pan now. Reduce the sauce in half or until it reaches the desired consistency. Swirl in a tablespoon of butter at the end to finish. Season to taste with salt.
9. Slice and serve the eye-of-rounds with the pan sauce.

Note: The measurements are for an average size deer. Double the ingredients if using elk.