To make the persimmon puree, peel and seed the persimmons, then blend with lemon juice in a food processer until smooth.
To make the flan, in a heavy-bottomed saucepan, bring the water and half a pound of the sugar to a boil. Cook until the caramel is smooth and has reached an amber color. Immediately spoon 2 tablespoons into the bottoms of ceramic ramekins. Place the ramekins in a deep roasting pan.
In another heavy-bottomed saucepan, bring the milk, cream, cinnamon, persimmon puree and vanilla to a boil. Remove from the heat, cover and let steep for about 10 minutes. Meanwhile, whisk together the eggs and egg yolks with the remaining sugar until thoroughly combined. Whisk a third of the milk mixture into the eggs, to temper them, then whisk the egg mixture into the remaining milk mixture. Strain through a fine-meshed sieve, and discard the vanilla bean.
Fill the ramekins three-quarters of the way full. Fill the roasting pan with cold water halfway up the sides of the ramekins. Bake uncovered in 350° oven (not convection) for 75 minutes, or until the custard jiggles only slightly in the center. Allow to chill completely. To serve, place a ramekin upside down in a shallow bowl, gently tap the bottom and lift it off the flan.