1. Place a rack in center of oven and preheat to 375°F. Line a large baking sheet with parchment.
2. Cut peaches in half and remove pits. Place a peach half cut side down on cutting board. Holding it together with one hand, cut 6 to 8 even slices through the stem end, then push gently to fan out. Combine sugar, nutmeg and pecans in a bowl.
3. Place one phyllo sheet on counter with narrow side facing you. Brush bottom half lightly with melted butter and sprinkle with sugar-pecan mixture. Fold top half over bottom half, pressing lightly. Repeat, folding top half over bottom half. Repeat again, this time brushing and sprinkling left half and folding right half over. Repeat once more. You should now have a 3-by-4-inch rectangle. Place on baking sheet. Repeat with remaining 3 phyllo sheets. Place a fanned peach half in center of each phyllo rectangle, brush with remaining butter and sprinkle with remaining sugar mixture.
4. Bake until phyllo is golden brown and crisp and peach has softened, 20 to 25 minutes, rotating pan halfway through. Place on a wire rack to cool.