Preheat oven to 350°
Crush gingersnaps to a fine crumb using a food processor. (Gingersnaps can also be crushed in a large zip-top bag with a rolling pin. Be sure to crush finely to yield the best crust.) Add melted butter and mix until well combined.
Press mixture into a 9-inch tart pan with removable sides. Make sure to press tightly onto the bottom and up the sides of the tart pan. Place pan on a baking sheet and bake at 350° for 10-11 minutes or until golden and fragrant. Remove from oven and let cool.
Combine heavy cream, milk, and 1/4 cup sugar in a medium saucepan. Combine remaining 1/2 cup sugar, cornstarch, and egg yolks in a medium bowl. Heat saucepan of milk mixture over medium-high heat until mixture just begins to simmer. Scoop about 1/4 cup hot milk out of saucepan with a ladle or cup; slowly drizzle warm milk into egg yolk mixture, stirring constantly with a wire whisk.
Slowly add egg mixture to remaining milk mixture in the saucepan, stirring constantly with a wire whisk. Heat mixture for 1-2 minutes, or until mixture thickens to a pudding consistency.
Remove from heat and stir in vanilla, cinnamon, ginger, and nutmeg. Transfer custard to cooled crust. Cover with plastic wrap covering the surface of the custard; chill for at least 1 hour or overnight.
Thinly slice persimmon horizontally to about 1/8-inch using a mandolin or very sharp knife. Cut each slice in half, to yield half circles. Roll one half circle into a tight roll. Place it upright, skin side up, on the tart and continue to arrange half circle concentrically around to make a rose shape. Repeat with remaining slices of persimmon. Skin may be tender, so work delicately. Use remaining scraps for another purpose.
Serve immediately or chill before serving. Store, covered, in the fridge.