Cut the peaches in half and remove the pit: With a sharp paring knife, slice each peach in half. Then, twist each half in opposite directions to pull the halves apart. Use your fingers to remove the pit. If it's stuck, carefully slide the knife under the pit and then pull it away.
Brush the peaches all over with melted butter.
Place the peaches, cut side down, on the grill. Grill them until tender and slightly charred, for about 4 minutes per side.
Serve immediately.
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Notes
If you don't own a grill, you can use a grill pan. Or, if you have a broiler-safe dish, broil the peaches for 3-4 minutes per side. Check on them often to make sure they don't burn. I suggest you start checking on them after 2 minutes. Please position the rack 6 inches below the flame, not directly below.
The best peaches to use in this recipe are ripe yet firm. Soft, very juicy peaches are best enjoyed raw since they won't withstand the cooking process and will turn mushy. However, you don't want them too firm, or they'll be impossible to halve.
You can keep the leftovers in a sealed container in the fridge for 3-4 days. They are delicious cold, or you can reheat them in the microwave at 50% power.