Combine 1 cup sugar and water in a small, heavy saucepan over medium-high heat. Cook until sugar dissolves; stir as needed to dissolve sugar evenly. Cook 9 minutes or until golden. Immediately pour into a 9-inch round cake pan, tipping quickly until sugar coats bottom of pan.
Place persimmon quarters in a food processor; process until smooth, scraping sides of bowl once.
Preheat oven to 350°.
Combine 1/2 cup sugar and flour. Beat cream cheese at medium speed of a mixer until smooth. Add flour mixture; beat until well-blended. Add egg whites and eggs; beat well. Gradually add milk and 1/2 cup persimmon puree; beat well. Pour batter into prepared cake pan. Place cake pan in a broiler pan; add hot water to broiler pan to a depth of 1 inch. Bake at 350° for 1 1/2 hours or until a knife inserted in center comes out clean. Remove cake pan from pan; cool completely on a wire rack. Cover and chill 8 hours.
Loosen edges of flan with a knife or rubber spatula. Place a serving plate, upside down, on top of pan; invert flan onto plate. Drizzle any remaining caramelized syrup over flan. Sprinkle with pomegranate seeds; garnish with persimmon slices.