Prepare the grill for direct cooking over high heat (230º to 290ºC). If you're using a charcoal grill, keep a small part of the charcoal grate clear of briquettes as a safety zone of indirect heat.
Pat the steaks dry, brush them with the oil, and season them evenly with the coarsely ground peppercorns, 1 tablespoon of the thyme, and 1 teaspoon kosher salt, pressing the spices into the meat. Allow the steaks to stand at room temperature for 15 to 30 minutes before grilling.
Related Grill Skills
Chopping Thyme
Meanwhile, in a large nonstick skillet on the side burner of the grill or on the stove, melt 2 tablespoons of the butter over medium-high heat. Add the shallots and cook until they begin to soften, 2 to 3 minutes. Add the remaining 1½ tablespoons thyme and stir for 10 seconds. Stir in the dried cherries and port (be careful of a flame up) and bring to a boil. Continue to boil until the liquid is reduced by about half, 6 to 8 minutes. Add the fresh cherries and broth and boil until the cherries are tender and about one-half of the liquid is cooked away, 6 to 8 minutes. Remove the skillet from the heat. Season the sauce with ¼ teaspoon salt and ¼ teaspoon pepper. Set aside.
Brush the cooking grates clean. Grill the steaks over direct high heat, with the lid closed, until cooked to your desired doneness, 7 to 9 minutes for medium rare, turning once or twice (if flare-ups occur, move the steaks temporarily over indirect heat). Remove the steaks from the grill and let rest for 3 to 5 minutes. While the steaks rest, finish the sauce.
Brush the cooking grates clean. Grill the steaks over high heat, with the lid closed, until cooked to your desired doneness, 7 to 9 minutes for medium rare, turning once or twice (if flare-ups occur, move the steaks temporarily over indirect heat). Remove the steaks from the grill and let rest for 3 to 5 minutes. While the steaks rest, finish the sauce.
Warm the sauce in the skillet over medium heat just until hot. Remove from the heat. Add remaining 3 tablespoons butter and whisk butter is absorbed into the sauce. Serve the steaks warm with the cherry sauce spooned on top.
