Ask your fishmonger to fillet the salmon and remove the backbone and small pin bones. Place one fillet, skin side down, in a deep glass, enamel, or stainless steel baking dish. Chop one bunch of dill and press into the flesh of the fish. In a separate bowl, mix together salt, sugar, and crushed pepper and cover the fillet with the mixture. Place the other fillet on top, skin side up, and wrap in plastic, and then in aluminum foil. Place a heavy platter on top of the fish and put 4 cans on top, distributing the weight of the cans evenly. Refrigerate for at least 48 hours, and up to 3 days. After the allotted time, remove the fish, and wipe off the dill and salt mixture, pat dry with paper towels.
Slice the salmon, on the bias, very thinly, and serve topped with the remaining dill sprigs, toast points and lemon wedges.