Cut the peaches in half along the natural seam and remove the pit. Try to maintain the beautiful half-moon shape that creates the perfect vessel for your toppings later.
In a small bowl, combine the melted butter, 1 tablespoon of honey, and brown sugar. This mixture will create a gorgeous caramelization on your peaches.
Heat your grill to medium-high (approximately 375-400°F). For gas grills, preheat for 10-15 minutes with the lid closed. For charcoal grills, wait until the coals have a light coating of ash.
Brush the cut sides of peaches with your butter-honey mixture. Place the peaches cut-side down directly on the grill grates. Grill for 3-4 minutes until distinct grill marks appear and the fruit begins to soften.
Flip and grill for another 2-3 minutes. You'll know they're done when the juices begin to bubble slightly.
While the peaches are grilling, mix the Greek yogurt with vanilla extract and the remaining 2 tablespoons of honey.
Place the warm grilled peaches on serving plates, cut-side up. Top each with a generous dollop of the honey-vanilla yogurt.
Drizzle with additional honey, sprinkle with cinnamon, and garnish with fresh mint and chopped nuts if desired. Serve immediately while the contrast between warm fruit and cool topping is at its peak.