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Scrub the outside of the shells with a scrub brush under cold, running water to remove all dirt. Discard any open or cracked shells. Let the oysters sit under water briefly, and then remove from the water while letting them drain. Use a charcoal grill or gas grill. Bring the grill to a medium high heat. Lay the oysters on the grill, flat side on the grill itself. Although there's little difference in method, how you cook your oysters will depend on whether you want to season your oysters beforehand or leave them unseasoned until right before you plop them down your gullet. If you want to season your oysters beforehand, shucking them is your best bet. If you'd prefer to season them after cooking, or perhaps not at all, you're better off leaving them in their shells while they cook. How do you shuck oysters? Wrap the top of the oyster in a towel or put on heavy-duty gloves for protection. Slide an oyster knife into the hinge (backside) of the oyster. Twist the oyster knife by turning your wrist, like you're turning on the ignition in your car with a key. Scrape the blade across the top part of the oyster shell, twisting to open the muscle. Remove the top portion of the shell and loosen the oyster foot from the bottom shell with your knife. Oysters are great raw or cooked in their own brine, but sometimes a little seasoning makes the oyster shine even more. Look for ideas that suit your own tastes. For inspiration try out some of the following:  Butter and garlic Butter and soy sauce Butter, shallot, fresh parsley, cheese (Pecorino), cayenne, paprika BBQ sauce Shut the lid to the grill and allow it to remain closed for 5 or 6 minutes. Open the lid and check on the oysters. What you're looking for will differ depending on how you prepared them:  Whole oysters should be checked for opening of the shell. You'll initially notice a line beginning to separate in the shell. Look for bubbling of the oyster brine inside that small separation. Throw out any oysters that haven't opened after 5-10 minutes. Oysters on the half shell need to be vetted before and during the shucking process to ensure they're safe to eat. If the oyster is already open prior to shucking, or it doesn't put up any resistance during shucking and pops right open, throw it out. Oysters on the half shell will tighten slightly when cooked; their liquid will bubble and help cook them over the course of 5-10 minutes. Serve with drawn butter, lemon, or as-is.

Summary:
Prepare the oysters for grilling. Prepare the grill. Decide whether to cook the oysters whole or on the half-shell. Prepare a seasoning for oysters on the half shell (optional). Cook the oysters. Remove whole oysters or oysters on the half-shell carefully so that they retain their juices.