Preheat oven to 350 degrees F (175 degrees C). Grease a 10x16-inch baking pan.

Combine flour, 2 1/2 cups sugar, persimmon pulp, milk, butter, eggs, baking powder, and baking soda in a large bowl; beat with an electric mixer on high speed until smooth, about 1 minute. Pour into the prepared pan.

Bake in the preheated oven until set, about 1 hour. Increase oven temperature to 400 degrees F (200 degrees C). Continue baking until top is browned, 10 to 15 minutes more. Let cool, about 10 minutes; pudding will sink in the center.

Combine 1 cup sugar, water, and vanilla extract in a small saucepan over medium heat. Cook until sugar has dissolved into a clear glaze, 3 to 5 minutes. Pour over warm pudding; let stand until set, about 10 minutes.