Preheat your oven: Preheat the oven to 375°F (190°C).
Prepare the puff pastry: Roll out the puff pastry sheet to fit your tart pan. Press it into the pan and trim any excess.
Prepare the peaches: In a bowl, combine the sliced peaches, granulated sugar, flour, ground cinnamon, and almond extract. Toss gently to coat the peach slices evenly.
Arrange the peaches: Arrange the peach slices in an overlapping pattern over the puff pastry, leaving a border around the edges.
Bake: Bake in the preheated oven for about 20-25 minutes or until the pastry is golden brown, and the peaches are tender.
Glaze the tart: While the tart is still warm, heat the apricot jam in a small saucepan until melted. Brush the melted jam over the peaches for a glossy finish.
Cool and Serve: Allow the tart to cool slightly before slicing. Serve your French peach tart at room temperature.
Optional: Serve with whipped cream or vanilla ice cream for an extra treat!