First, cream the room-temperature butter and powdered sugar together in a stand mixer on low. Add the egg and mix; don't worry if it looks curdled. Add the dry ingredients in two parts, mixing just until a dough forms.
Shape the dough into a flat, 6-inch disk, wrap it tightly in plastic, and let it chill in the refrigerator for at least 30 to 60 minutes. This step is crucial for making the dough easier to roll out later.
Preheat your oven to 375°F (190°C). Peel the peaches, cut them in half to remove the pits, and then slice each half into three even wedges. Set them aside while you prepare the caramel.
In a 10-inch oven-safe skillet, melt the sugar over medium-low heat without stirring. Swirl the pan as it liquefies. Once melted, add the butter and stir until it forms a smooth caramel. Cook for another minute or two.
Remove the skillet from the heat, stir in the cinnamon and vanilla paste, and let it cool slightly. Arrange the peach slices decoratively in the pan. Return to the stove, add the water, and simmer covered for 5 minutes.
Remove the lid and continue to cook for 15-20 minutes, allowing the liquid to reduce until it's almost completely gone. This prevents a soggy crust. While it reduces, roll your chilled dough into a 12-inch circle.
Place the rolled dough over the peaches in the skillet, tucking the edges down around the fruit. Cut about five slits in the top for ventilation. Bake for 25-30 minutes, or until the crust is golden brown.
Let the tarte cool in the pan for 30 minutes. This allows the caramel to set slightly. Place a large serving plate over the skillet and, using oven mitts, carefully and quickly flip it over to invert the tarte.