Mix soy sauce, vegetable oil, wine vinegar and paprika in a square container with a liquid-tight lid large enough to hold all of the tofu. Drain water from tofu and wrap in clean towels, press carefully to soak up water without breaking the tofu. Unwrap tofu and slice into 5 thick slices per block. Place the slices in the marinade, put the lid on the container and turn it over to coat the tofu. Refrigerate the marinating tofu for at least 24 hours or up to three days, turning occasionally to coat.
Prepare the grill for smoking (see Tips & Notes for instructions).
When the grill is ready and the cool side of the grate is oiled, place the tofu on the grate. Close the lid, opening it once every 10 minutes to quickly turn the tofu. Smoke tofu for 40 minutes.
Serve on buns with lettuce, tomato, and condiments of choice.