Position a rack in the center of the oven and heat the oven to 425°F. Smear the butter over the bottom of a 9x13-inch baking dish.
Using a vegetable peeler in a short zigzag motion, peel the peaches from top to bottom. Halve and pit the peaches. Sprinkle the cut sides with the nutmeg.
In a small saucepan, bring the maple syrup, salt, and 3/4 cup water to a simmer over low heat, stirring occasionally.
Meanwhile, on a floured work surface, pat the dough into a 4x6-inch rectangle. Using a floured rolling pin, roll the dough into a 12x18-inch rectangle, lightly dusting with more flour to keep dough from sticking, if necessary. Cut the dough into 6 squares.
Place a peach half cut side down in the center of each piece of dough. Pull the four corners of each piece of dough up and over each peach, pinching the edges tightly to seal. Transfer the dumplings to the prepared dish in a single layer in two rows, about 1/2 inch apart.
Pour the hot syrup around the dumplings. Brush the dumplings with the milk and sprinkle with the sugar. Bake until the dumplings are golden-brown, about 25 minutes.
Let cool in the baking dish on a wire rack until warm, about 15 minutes, before serving with crème fraîche, sour cream, or yogurt, if using.