Preheat your outdoor gas grill to medium heat (375-400°F / 190-204°C). If using a charcoal grill, light the coals and let them burn until covered with white ash, then spread them evenly. For an indoor cast iron grill pan, preheat it over medium heat for 3-4 minutes.
In a small mixing bowl, combine the melted butter, brown sugar, cinnamon, and vanilla extract. Stir well until the sugar is mostly dissolved and everything is evenly incorporated.
Wash the peaches and pat them dry. Using a sharp chef's knife, cut each peach in half along the natural seam and twist to separate. Remove the pit (easily pops out for freestone peaches; carefully dig out with a spoon or paring knife for clingstone).
Using a pastry brush, generously coat the cut (flat) side of each peach half with the butter mixture, making sure to cover the entire surface including the rim and pit cavity.
Place each peach half cut-side down on the preheated grill grates or grill pan. Cook undisturbed for 3 to 5 minutes until you see clear, defined grill marks and lightly golden edges.
Using tongs, gently flip each peach half skin-side down and grill for an additional 2 to 3 minutes until tender but still holding their shape. Add a pinch of sea salt if desired, then remove from the grill.
Place 1-2 grilled peach halves on each serving plate, add a generous scoop of vanilla ice cream, drizzle with honey, and garnish with fresh mint leaves. Serve immediately while warm.