1. Prepare the grill for direct heat. If using a gas grill, preheat to medium high (450℉). If using a charcoal grill, light the charcoal or wood briquettes; when the briquettes are ready distribute them evenly under the cooking area. For a medium hot fire you should be able to hold your hand 6'' above the coals for about 4-5 minutes.
2. Place the grill rack on the grill. Meanwhile, melt the butter in a small saucepan over a low heat. Stir in the peach pieces and the Old Bay Seasoning.
3. Cook for about 3 minutes, stirring frequently to attain a thick, emulsified texture. Rinse the oysters, scrubbing them with a stiff-bristle brush to remove any grit.
4.Place the oysters on the grill. Once their shells have opened about 1/4 inch remove them from the grill.
5. Use a small knife to pull off the top half of each shell; discard those shells. Return the bottom halves (with the oyster and remaining liquid) to the grill and top each oyster with a teaspoon of the Old Bay Seasoning-peach sauce.
6. Grill for 1-2 minutes, until they bubble.
7. Arrange them on a platter. Eat directly from the shell.