Cut the teres major steak into 1"-2" rounds that resemble filet mignon.

Melt half of the butter in a medium saucepan pan over medium heat. Add shallots and mushrooms; sauté for 4 minutes. Add half of the wine and half of the broth; cook for 5 minutes, stirring frequently. Set aside.

Meanwhile, cook the steaks. Sprinkle pepper over steaks. Melt remaining butter in a safe nonstick pan or cast iron skillet over medium-high heat. When bubbling, add steaks; cook 3 minutes on each side. Reduce heat to medium-low; cook 1 1/2 minutes on each side or until desired degree of doneness. Place on a platter; keep warm.

Add remaining wine and broth to skillet. Whisk in coconut aminos and arrowroot to thicken. Cook 1 minute, whisking, until arrowroot is dissolved. Add reserved mushroom mixture and thyme, cook 1 minute.

Serve sauce with steaks.