Unroll the puff pastry and cut into pieces, if you wish. Use a knife to score a 1cm border around the edge without cutting through the pastry completely. Put the pastry on the lined baking sheet.
Prick the pastry all over inside the border, brush with the beaten egg and bake for 12-15 minutes, then remove and push down the base inside the border and set aside to cool.
In a bowl, mix the crème fraîché, eggs and yolk and the Parmesan together, lightly season, then spoon the filling onto the cooled pastry, ensuring that none goes over the border.
Arrange the asparagus tips and the strawberries over the mixture and top with a little more cheese (if using) or, using a pastry brush, brush with a little of the beaten egg.
Bake for about 20-25 minutes until the pastry is golden brown and the filling is set. If the tarts are small, they will probably be cooked in just 10 minutes.