Breakfast Bread Pudding with Peaches and Vanilla-Buttermilk Syrup

To make peach sauce: In a medium saucepan over medium-high heat, bring sugar and 1/2 cup water to a simmer, stirring until sugar dissolves. Reduce heat to medium-low; stir in vanilla bean and lemon juice. Add peaches and simmer gently until softened, 10 to 12 minutes. Let cool slightly, remove vanilla, then use a slotted spoon to transfer half of peach mixture to bowl of a food processor. Pulse 4 to 5 times, until mixture is a chunky sauce; set aside for assembly. Reserve the rest of peaches and their juices in a dish, and refrigerate until ready to serve. Preheat oven to 350 degrees F. Toast bread slices in a single layer on a rimmed sheet tray until just golden and dried out, 8 to 10 minutes. Flip slices and toast 5 to 7 minutes more. To make custard: Meanwhile, in a medium saucepan, combine milk, cream, sugar, vanilla bean, nutmeg, and salt. Bring to a simmer over medium heat. Remove from heat, cover, and let steep 20 minutes. Remove vanilla. Whisk eggs and yolks in a medium bowl, then whisk into steeped milk-cream mixture. To assemble: Grease a deep pie dish. Arrange 3 to 4 pieces of bread on bottom of dish, spread 1/2 cup peach sauce on bread layer, and repeat with remaining bread slices and peach sauce, ending with bread slices. Slowly pour custard into dish, gently pressing down on bread as needed so it absorbs custard (pudding may need to sit a few minutes before adding more custard). Discard any leftover custard. Cover and let sit at least 40 minutes or overnight in the refrigerator. Preheat oven to 350 degrees F. Sprinkle top of pudding with 2 tablespoons sugar and garnish with a few gratings of nutmeg. Bake until golden, puffed, and set, 45 to 50 minutes. Transfer to a rack and let sit 20 minutes. To make vanilla-buttermilk syrup: In a medium saucepan, cook cream, buttermilk, sugar, and cornstarch over medium heat, stirring, until sugar melts. Increase heat to medium-high and bring to a boil, stirring occasionally, 2 minutes more. Remove from heat and whisk in baking soda, vanilla bean, and salt; mixture will foam and froth. Let sit 2 minutes. Serve pudding immediately with reserved peaches and vanilla-buttermilk syrup.