Clean the grill thoroughly and oil the grates to prevent sticking. Dip a folded paper towel into cooking oil and rub it along the grates with grill tongs. Then preheat it to medium-high heat or around 375-400°F (190-205℃). If using charcoal, make sure the coals are evenly lit and glowing.
Mix the water and lemon juice (to keep the cut apples from turning brown) in a large bowl and set aside while prepping the apples.
Wash the apples, removing stickers and the stems, and core them. Cut them into thick slices or wedges. Or, if using small apples, cut them in half lengthwise.
Place the sliced apples in the lemon-water mixture and mix well. If the apples float, you can keep them submerged with a heavy plate.
Melt the butter and set it aside.
Take the apple slices out of the lemon water, shake off excess liquid, and pat them dry. Then brush them with the melted butter. If doing a savory version, sprinkle them with a little salt.
Arrange the apple slices on the grill grates, and grill them for 2-3 minutes per side or until they have gorgeous grill marks. They should soften slightly but still be firm. Place them on a serving platter or individual plates.
Optionally, mix the brown sugar with the cinnamon, allspice, and salt in a small bowl (adjust amounts to taste). Set it aside.
If using the sugar and spices, sprinkle them with the cinnamon sugar on both sides until evenly coated, and serve them warm with ice cream.