In a large bowl combine the flour, granulated sugar, baking powder, and salt and blend well. Add the cold butter and mix with a fork or a pastry blender just until the mixture resembles coarse bread crumbs. Add the half-and-half and gently stir it in (the mixture will be crumbly). Then use your hands to mix the dough quickly and gently in the bowl just until it comes together. Turn the dough out onto a lightly floured work surface. Lightly dust your hands with flour and gently pat out the dough to about ¾ inch thick. Dip a 2½-inch round biscuit cutter in flour and cut out rounds of dough. Gather scraps of dough and pat out, using a light touch so you don’t overwork the dough; make a total of eight shortcakes. Place the shortcakes about 2 inches apart on a baking sheet lined with parchment paper. Brush the tops with the melted butter. Bake the shortcakes in the oven until golden brown, 15 to 20 minutes. Set aside to cool.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
In a large bowl whip the cream, confectioners' sugar, and vanilla just to stiff peaks; do not overbeat. Cover and refrigerate until serving.
Brush the cooking grates clean. Sprinkle the cut side of the peach halves with the brown sugar. Grill the peach halves, cut side up, over direct medium heat, with the lid closed, until the sugar melts, the peaches are tender, and the skins are charred, 8 to 10 minutes. Carefully transfer the peaches to a platter, pouring the juices from the pits into a medium bowl.
Pull the charred skin off the peaches and discard. Cut the peaches into bite-sized pieces and add them to the bowl with the juices. Gently toss to coat. Split each shortcake horizontally and top each bottom half with equal portions of the peaches and whipped cream. Add the shortcake tops and garnish with mint sprigs, if desired.