Place applewood chips in center of 1 (18-inch) square piece of heavy-duty aluminum foil. Bring up sides of foil over wood chips; double fold top and side edges to seal, making a packet. Pierce several holes in top of packet.
Preheat a gas grill to medium (350°F to 400°F) on one side, or push hot coals to one side of a charcoal grill. Place applewood smoke packet, with holes facing up, directly on lit burner or hot coals.
Line a baking sheet with heavy-duty aluminum foil. Gently stir together jumbo crabmeat and crab claw meat; spread mixture on prepared baking sheet.
Place baking sheet over unlit side of grill. Grill crabmeat, covered, 30 minutes. Remove crabmeat, and cool completely on baking sheet, about 15 minutes.
Stir together crushed crackers, Dijon mustard, mayonnaise, lemon juice, melted butter, Worcestershire sauce, Old Bay seasoning, parsley, dry mustard, and eggs in a large bowl until well combined. Gently fold in smoked crabmeat, and shape mixture into 8 (1-inch-thick) crab cakes.
Heat 2 tablespoons of the oil in a large skillet over medium. Add 4 crab cakes, and cook until golden, 4 to 5 minutes on each side. Repeat with remaining oil and crab cakes. Drain on paper towels. Serve immediately with Tartar Sauce.
Pierpoint Restaurant, Baltimore, MD