Prepare the grill for direct cooking over high heat (450° to 550°F).
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.
Meanwhile, in a large nonstick skillet on the side burner of the grill or on the stovetop, melt 2 tablespoons of the butter over medium-high heat. Add the shallots and sauté until they begin to soften, 4 to 6 minutes. Add the remaining 1½ tablespoons thyme and stir for 10 seconds. Stir in the dried cherries and port (watch for 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 as much as possible, 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.
Place the skillet over medium heat and add the remaining 3 tablespoons butter. Stir until the butter is melted. Remove from the heat. Serve the steaks warm with the cherry sauce spooned on top.