Preheat oven to 400°F.
Combine flour, white sugar, baking powder and salt in food processor; pulse to combine. Add 4 tbsp cold butter, cut in small cubes, and pulse 6-8 more times until it resembles coarse meal. Transfer to a bowl.
Whisk egg and milk until combined. Pour over flour mixture. Stir with a big spoon until well-combined and forming a sticky dough. Turn out onto a heavily-floured surface and knead until soft but workable. Wrap in plastic and refrigerate.
Generously butter a 9" pie plate. Turn a Pyrex or ceramic ramekin upside down in the middle of the plate and butter the outside well too. Cut peaches in half and remove pits. Place peach halves back together and arrange around the ramekin.
In a small saucepan, combine brown sugar, salt, water and vanilla with remaining 3 tbsp butter. Heat over medium heat until sugar is dissolved and butter is melted. Pour syrup evenly over peaches.
Roll dough out to 11" circle. Drape dough over peaches, tucking dough into the pie plate; do not crimp dough to edge of dish. Place on a rimmed cookie sheet and place in oven for 25-35 minutes until biscuit is golden brown. Immediately transfer to a rack and let cool 30 minutes.
Place a large platter over the dish and quickly invert. Most of the syrup should be in the cup. To serve, slice so each diner gets a whole peach, and drizzle with remaining syrup. Serve with whipped cream or vanilla ice cream.