Swish mushrooms in a bowl of water to clean, then trim any tough stem ends. Cut into halves, or quarter if large. Combine salt and pepper in a small bowl. Sprinkle steaks all over with 2 tsp. thyme leaves and half of salt-and-pepper mixture, pressing seasonings into meat.
Heat 2 tbsp. oil in a large cast-iron skillet on a camp stove over medium-high heat. Add mushrooms, thyme sprigs, and remaining salt mixture. Cook, stirring often, until mushrooms are tender and browned, 14 to 20 minutes. Transfer mushrooms and thyme to a bowl; tent with foil.
Heat remaining 1 tbsp. oil in pan over medium-high heat. Cook steaks on both sides, turning once, until browned and done the way you like (cut to test), 8 to 14 minutes total for medium-rare. Transfer steaks to a cutting board and tent with foil.
Discard fat from pan. Pour wine into pan and cook until reduced to 2 to 3 tbsp. (don't reduce further or sauce may separate), 1 to 2 minutes. Whisk in butter until melted and blended. Cut steaks in half and set on plates. Drizzle with sauce and serve with mushrooms.