Preheat the oven: Preheat the oven to 375°F (190°C).
Prepare the peaches: Wash and dry the peaches. Slice each peach in half around the natural seam, twist the halves apart, and remove the pits.
Melt the butter: Add the butter to a medium baking dish. Place the dish in the oven for 1 to 2 minutes, just until the butter melts.
Add the flavorings: Remove the dish carefully. Stir the honey, cinnamon, and nutmeg into the melted butter until combined.
Arrange the peaches: Place the peach halves in the baking dish with the cut sides facing up.
Coat with the sauce: Spoon the honey butter over the peaches, making sure some of the mixture collects in the centers.
Bake: Bake uncovered for 20 to 25 minutes, or until the peaches are tender when pierced with a small knife.
Baste the peaches: Halfway through baking, carefully spoon some of the sauce from the bottom of the dish over the peaches.
Check the texture: The peaches should be soft enough to cut with a spoon but still hold their shape.
Rest: Remove the baking dish from the oven and let the peaches rest for 5 minutes.
Finish with the sauce: Spoon the warm honey-cinnamon sauce over the peaches again.
Serve: Serve warm, with one or two peach halves per person. Add vanilla ice cream and extra sauce when desired.