For the spiced tart crust, mix together the flour, sugar, cinnamon, and ginger in a large bowl. Grate the frozen butter over the bowl, pausing to mix in the butter shards every 20 seconds so they don’t just form a large butter clump on the top of the bowl. Add the cold milk and then the water, 1 tablespoon at a time, stirring the dough with a wooden spoon and then working it slightly with your hands to incorporate all the flour. If it doesn’t come together, add another tablespoon of cold water and mix, repeat if necessary until a solid dough forms. Wrap with plastic wrap and refrigerate for 30 minutes before rolling it out until it is about 1/4 inch thick, making a circular shape that is at least 11 inches in diameter. Press it into a 9-inch tart pan and trim off the excess. Prick it all over with a fork and then cover with plastic wrap and place in the freezer for at least 30 minutes.For the custard, preheat the oven to 350 degrees Fahrenheit. Fill a large casserole dish 3-inches deep with water and place it on the lowest rack in the oven.

Whisk together the eggs and yolk in a medium bowl until combined. Set aside.

In a small saucepan, whisk together the creme fraiche, 1/3 cup sugar, milk, flour, cinnamon, cloves, nutmeg, and salt over low heat until combined. Allow to cook until hot but not boiling, stirring every minute. Remove from heat and ladle one scoop of the creme fraiche mixture into the egg mixture and whisk vigorously. Continue ladling the hot creme fraiche mixture into the egg mixture until all of the mixture has been incorporated, whisking constantly. Stir in the fresh rosemary and set aside.

Remove the tart crust from the refrigerator and remove and discard the plastic wrap. Pour the custard into the tart crust until the tart is nearly full. Arrange the persimmon slices in four small fans, one in each quarter of the tart, and place one slice in the center. Sprinkle the top of the tart with the remaining 2 tablespoons granulated sugar. Place in the center rack of the oven and bake until the custard is a deep golden brown and has just begun to set, about 45 minutes. Turn the oven off, leave the oven door ajar, and allow to cool for 30 minutes in the oven before removing the tart from the pan.

While it is cooling, heat the honey and water in a small saucepan over low heat until they form a smooth syrup, about 4 minutes, stirring every minute. Remove from heat.

Once the tart has cooled, use a pastry brush to brush the persimmons with the honey syrup. Serve immediately.