Preheat the oven to 400°F. Cover a large baking sheet with parchment paper.
In a small bowl, stir together the cream cheese, brown sugar, vanilla extract and apple pie spice. Set aside.
Cut off the tops (stems) of persimmons and thinly slice vertically. Slices should be about 1/8 of an inch thick. Colorful skins can stay on.
Roll out puff pastry directly onto parchment paper and gently pull ends to make a rustic circle or square. Fold up edges to form a rim. This doesn't need to be perfect. Spread the cream cheese mixture in the center of the pastry.
Arrange the persimmon slices over the cream cheese mixture.
Baste the pastry edges and top of the persimmons lightly with cream. Sprinkle 1 tablespoon of the turbinado sugar over the pastry and persimmon slices.
Bake for 15-18 minutes or until edges of pastry are lightly browned. It might be a little puffy, this is fine.
Remove and allow to cool before making and drizzling with icing.
Maple Icing:
In a medium bowl, whisk together the powdered sugar, maple syrup and heavy cream until smooth. Drizzle over the pastry. Top with the remaining 1/2 tablespoons of turbinado sugar while still tacky.
Slice and serve with a scoop of vanilla bean ice cream.
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