Let persimmons stand at room temperature about 1 hour if they have been refrigerated. Preheat oven to 350 F. Grease and flour a 9x3-inch springform pan; set aside.
Peel persimmons or scrape pulp from peel with large metal spoon. Press persimmons through a single mesh, large-gauge (coarse mesh) strainer until you have 1 cup of strained pulp. In a small bowl combine the 1 tablespoon flour and the brown sugar; stir in strained pulp or applesauce. Set aside.
For streusel mixture, in a medium bowl stir together the 1/2 cup flour, 1/2 cup granulated sugar, and cinnamon. Cut in 1/4 cup butter until mixture resembles coarse crumbs. Stir in the walnuts; set aside.
In a medium bowl stir together the 2 cups flour, baking powder, baking soda, and salt; set aside. In a large mixing bowl beat the 1/2 cup butter with an electric mixer on medium to high speed for 30 seconds. Beat in the 1 cup sugar and vanilla. Add eggs, one at a time, beating after each addition just until combined. Add flour mixture and sour cream alternately to beaten mixture, beating just until combined after each addition. (Batter will be stiff.)
Spread half of the batter evenly into the bottom of the prepared pan, building up a 1-inch rim of batter around edges of pan. Spoon persimmon mixture into center of pan. Carefully spoon remaining batter in small mounds, covering top of persimmon mixture. Sprinkle streusel mixture evenly over batter.
Bake in preheated oven for 1 hour or until toothpick inserted into cake near center comes out clean. Cool in pan on a wire rack for 10 minutes. Loosen and remove sides. Cool completely on the wire rack. (Filling will sink somewhat as coffee cake bakes). Makes 12 servings.