Remove the venison steak from the refrigerator half an hour before cooking to bring it to room temperature. Cut the fresh red cabbage into thin strips and coarsely chop the peeled chestnuts.
Heat some butter or oil in a braising pan. Add the red cabbage with a small splash of water, turn the heat to low and cover the pan. Let the red cabbage braise gently for about thirty minutes until tender, stirring regularly.
Pat the venison steak dry with paper towels. Heat a skillet over high heat with butter and quickly sear the meat. Cook the steak approximately two to three minutes per side, depending on the desired doneness. Remove the meat from the pan and let it rest for a few minutes under aluminum foil.
Divide the warm braised red cabbage over the plates. Slice the venison steak and place it on top of the cabbage. Finally, garnish the dish by sprinkling the chopped chestnuts directly over the meat and vegetables.