Wash the peaches thoroughly under cold running water. Using a sharp knife, cut each peach in half along the natural seam that circles the fruit. Twist the halves in opposite directions to separate them, then remove the pit.
In a small bowl, combine the melted butter, brown sugar, cinnamon, vanilla extract, and salt. Stir until the sugar begins to dissolve into the butter.
Preheat your grill to medium-high heat (around 375-400°F). Clean the grates well and oil them lightly to prevent sticking.
Brush the cut sides of the peaches with the sweet butter mixture. Place the peaches cut-side down on the preheated grill.
Let them cook undisturbed for 3-4 minutes until grill marks form and the edges begin to caramelize.
Flip the peaches carefully, brush with remaining butter mixture, and grill for another 2-3 minutes until tender but not mushy.
Transfer the warm peaches to serving plates. Add a generous dollop of Greek yogurt beside each peach half, drizzle with honey, and sprinkle with your choice of optional garnishes.
Serve immediately while the contrast between the warm, caramelized peaches and cool, creamy yogurt is at its most delightful.