1. Peel peaches, remove pits, and slice into a heavy saucepan, catching all the juices. Stir in sugar, nutmeg, and cinnamon, and cook over medium heat until bubbly. Continue cooking, stirring occasionally, until the sauce reaches a syrupy consistency, about 10 minutes. Remove from heat.
2. Whisk together the eggs, milk, a pinch of cinnamon, and vanilla in a shallow bowl. Melt butter with oil over medium heat in a large nonstick skillet. Dip each piece of bread into the batter, add to the skillet, and cook until browned on both sides. Serve hot with the warm peach sauce. Top with confectioners' sugar.