Preheat your pellet grill or smoker to 350°F with your choice of wood or pellets.
For the canned claw or lump crabmeat - Drain the crab meat thoroughly. Pick through it carefully to remove any shells. Using a knife, roughly chop the crab meat. For the imitation crab - Use a knife to shred or chunk the meat. Mix the imitation and regular crab together and set aside.
In a large mixing bowl, combine the cream cheese, mayonnaise, and sour cream. Mix well until the cream cheese is fully incorporated and the mixture is smooth.
Add the minced garlic, lemon juice, black pepper, Worcestershire sauce, Cajun seasoning, and 1 tablespoon of the chives to the cream cheese mixture. Stir until well combined.
Fold in about 6 ounces of the grated Monterey Jack cheese and half of the Parmesan cheese to the mixture, reserving the rest of the cheese for topping.
Gently fold in the imitation crab and claw crab meat. If using lump crab, fold carefully to avoid breaking up the chunks.
Transfer the mixture to a grill-safe cast iron skillet or baking dish, spreading into an even layer. Then place the dish on the preheated smoker and cook for 20 minutes.
While the dip smokes, combine the reserved Monterey Jack cheese, reserved Parmesan cheese, and about ½ tablespoon of diced chives in a small bowl.
After 20 minutes, remove the dip from the smoker. Sprinkle the reserved cheese mixture over the top. Return the dip to the smoker and cook for an additional 10-12 minutes, until the cheese is melted.
Remove the dip from the smoker and let cool for 2-3 minutes before serving. Serve hot with crostini, crackers, corn chips, or vegetable crudités.