Carefully add antelope steak to the hot skillet and let it sizzle for about 2 minutes, then use tongs to flip it over and cook about 2 more minutes. Press down gently to ensure even contact between steak and pan. If your pan gets too hot, add a Tablespoon of cold oil to bring the temperature back down so you don't burn the antelope steak.
Continue to cook the antelope steak until it has developed a dark brown crust on both sides, and reached an internal temperature of 130 degrees F (for medium-rare) using an instant read meat thermometer inserted in thickest portion of meat.
Transfer steak to a cutting board, sprinkle both sides with ¼ teaspoon each rosemary and thyme, and let it rest for 10 minutes.
Meanwhile, in the same cast iron skillet over medium heat, add oyster mushrooms and butter with remaining ¼ teaspoon of rosemary and ¼ teaspoon of thyme and sauté, stirring occasionally, until tender, about 4 to 5 minutes.
Cut steak diagonally against the grain into ¼-inch-thick slices and divide among four dinner plates. Top the steak with some sauteed oyster mushrooms and season with a touch of additional kosher salt and freshly ground black pepper if desired. Serve immediately.