Mix salt, sugar, pepper, and cayenne for dry rub together and coat beef well on both sides.
Place on a rack set on a sheet pan and refrigerate, uncovered, for 12 to 24 hours, flipping over halfway through.
Mix barbecue sauce, vinegar, mustard, turmeric, and garlic together for wet rub and reserve.
Transfer beef to a dish and pour 1/2 of the wet rub over top. Use two forks to poke the surface all over for about 1 minute. Turn beef over and repeat with remaining wet rub. Spoon any excess on the bottom over the top.
Cover and refrigerate for 4 to 12 hours.
Preheat a charcoal grill until the briquettes turn white and ashy. Remove meat from the refrigerator, uncover, and scrape off excess wet rub.
Grill over hot charcoal until blackened and an instant-read thermometer inserted into the center of the meat reads 120 to 125 degrees F (49 to 52 degrees C), 7 to 10 minutes per side, or to desired doneness. Remove to a foil-lined plate, and quickly wrap up tightly in foil.
Let rest for at least 15 minutes, during which time the meat will finish cooking, and end up close to medium-rare.
Slice very thinly, toss in the meat drippings and juices left in the foil, and serve on buns.