Heat your grill to medium, around 375°F to 400°F (190°C to 200°C). Scrub the grates well with a grill brush once hot, then wipe them with an oil-dampened paper towel held in tongs.
Run a sharp knife along the natural seam of each peach, cutting all the way around down to the pit. Twist the halves apart and remove the pit, leaving the skin on.
In a small bowl, stir together the melted butter, brown sugar, cinnamon, vanilla, and pinch of salt until the sugar dissolves into a loose, pourable glaze.
Using the basting brush, coat the cut face of each peach half with the butter glaze, getting right into the pit well. A light coat on the skin side helps prevent sticking too.
Place the peaches cut-side down directly over the heat and leave them completely undisturbed for 4 to 5 minutes, until they release from the grates on their own and show deep grill marks.
Turn each half onto its skin side with tongs and grill for another 2 to 3 minutes, until tender with a little give but still holding their shape.
Move the peaches to a plate, cut-side up, and drizzle each half with honey. Add a scoop of vanilla ice cream to the warm pit well if serving as dessert, and serve right away.