Whisk yolks, sugar and flour, and vanilla in bowl. Add boiling milk while whisking continuously. Return to saucepan, and whisk over medium heat until thick and bubbly.
Transfer pastry cream to bowl, cover with plastic wrap directly onto surface of pastry. Chill. (can be made 2 days ahead.)
In a medium heavy saucepan, stir cup water and sugar. Heat over high heat until caramel forms, occasionally swirling the pan. Meanwhile, bring remaining 1 cup water to boil in a small saucepan. Remove caramel from heat. Mix in boiling water. Stir to dissolve caramel. Cool syrup.
Preheat oven to 400 degrees. Roll out puff pastry sheet on floured surface. Cut out nine 4-inch rounds. Arrange on a baking sheet. Pierce pastries several times with tines of fork. Chill 15 minutes.
Beat egg and milk to blend. Brush 6 pastries with egg wash; sprinkle with sugar. Bake pastries until puffed and golden. Transfer to rack and cool.
Cut each pastry in half horizontally. Reserve 6 sugar-coated tops.
Place 1 plain pastry round on each of six plates. Spread a tablespoon of pastry cream over. Top with sliced berries. Cover with second pastry. Repeat layering of pastry cream and berries. Top with reserved sugar-coated pastries.
Spoon caramel syrup around base of each napoleon.