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.
Splenda® or another sugar substitute may be used. Increase sweetener amount by 1/2 cup to make up for the difference in sweetness. Honey may also be used in place of some of the sugar to make it more like the original recipe was 100 years ago.
If you do not have easy access to American Persimmon pulp, Asian persimmons may be substituted. Pumpkin pulp may also be used.
Substitute margarine for the butter if desired.
Per Serving: 481 calories;9.4 g fat;
96.6 g carbohydrates;
5.3 g protein;
53 mg cholesterol;
209 mg sodium.