Preheat oven to 350°F. Sprinkle steaks on both sides evenly with salt and pepper. Let stand 10 minutes.
Heat oil in a large cast-iron skillet over medium-high. Add steaks; cook until deep brown, about 4 minutes. Flip steaks; cook until browned on other side, about 3 minutes. Transfer skillet to preheated oven; bake until a thermometer inserted in thickest portion of steaks registers 145°F for medium doneness, about 5 minutes. Remove from oven. Transfer steaks to a plate; let rest 5 minutes.
Meanwhile, whisk together yogurt, mustard, horseradish, and chopped thyme until fully incorporated.
To serve, spoon yogurt sauce evenly among steaks. Garnish with thyme sprigs.