Wash and scrub the oysters in cold running water, removing any sand, mud or loose shell. Place the clean oysters in a big bowl and keep refrigerated until ready to use (the same day)
Melt the butter in a small saucepan. Add the garlic and cook for 1-2 minutes without browning the garlic. Salt is not needed because the oysters are already salty.
Heat a grill to high heat. I use propane but charcoal grilling works well too. Remove oysters from the bowl and place on the grill using tongs. Discard any oysters that are already open. You want them to be tightly closed. Place them on the grill with the bowl-shaped side of the shell down. Grill the oysters until they pop open a bit. Remove from the heat with the tongs, and use an oyster shucker to pry and twist the oyster apart. Careful, it's HOT! Shucking oysters can be dangerous so never use a sharp knife. Use heat proof gloves or potholders to handle the oysters. . Once the oyster is open, scrape the flesh away from the muscle that attaches it to the shell. Scrape off the muscle and discard if it is on the bowl-shaped side of the shell. Place the oyster flesh in the bowl-shaped shell and set it aside. Discard the other side of the shell.
When all the oysters are open and ready to go, place them, in their shells, back on the grill with the tongs. Pour the butter mixture over the oysters with a spoon, dividing it equally. Continue cooking the oysters until their "liquor" has burned off, leaving just the butter. They should be lightly golden brown. Don't overcook them or they will be rubbery. Be careful when cooking the oysters with the butter. There is a lot of spitting and sputtering, but that adds a nice seared flavor. Add a little grated Parmesan cheese just before removing from the grill.
Remove the oysters with the tongs to a serving platter. I like to serve them nestled into a bed of coarse kosher salt. Enjoy piping hot with a dash of hot sauce if you like.