This version of classic tarte Tatin is made with fresh peaches and is topped with caramel sauce and sugared almonds or pistachios.
Heat oven to 375 degrees. Bring a medium saucepan of water to a boil. Blanch peaches in boiling water 15 to 20 seconds; remove with a slotted spoon. Use a paring knife to peel and cut peaches in half. Remove pits, and reserve four peach halves. Quarter remaining peach halves.
Butter four 5-inch tart pans with removable bottoms; dust with sugar. Arrange eight peach segments around edge of each pan, and place a reserved peach half cut side up in the center. Loosely wrap bottoms of pans with foil; place on baking sheet. Bake until peaches are tender and sugar and juices start to caramelize, about 30 minutes. Transfer to a wire rack to cool 10 minutes.
Cover tart pans with a layer of plastic wrap, and place another baking sheet on top to gently flatten peaches. Place filled cans or other weights on top of the baking sheet, and transfer to refrigerator to chill for at least 1 hour.
Raise oven temperature to 400 degrees. On a lightly floured surface, roll puff pastry to 1/8 inch thick, and cut out four 5 1/2-inch circles. Place rounds on buttered rimmed baking sheet, and transfer to freezer for 10 minutes. Remove from freezer, and place another baking sheet, buttered on the bottom, on top of the pastry rounds. Bake until golden brown and crisp, about 30 minutes. Transfer to wire rack to cool.
Make caramel sauce: Combine remaining 3/4 cup sugar and 3 tablespoons butter with the water in a small saucepan. Cook over medium-high heat, gently swirling pan occasionally to evenly color, until caramel is a medium amber color, 6 to 8 minutes. Wash down sides of pan with a pastry brush dipped in water to prevent sugar crystals from forming. Remove from heat, and slowly add heavy cream, stirring to combine. Be careful when adding cream to the caramel, as it will bubble vigorously. Set aside.
Remove tarts from refrigerator, and place a pastry crust on top of each. Invert onto serving plates; pour about 3 tablespoons caramel sauce over each. Garnish with pistachios or sugared almonds, if desired. Serve immediately.
The peaches may be baked and stored in the refrigerator up to six hours in advance.