In a 2qt saucepan, combine the strawberries, sugar, and lemon juice and bring the mixture to a boil over medium heat. Reduce the heat to medium-low and simmer for 8-9 minutes until the mixture becomes thick and syrup-like. Remove from the heat and mash the mixture with a potato masher or the back of a spoon to make the strawberry pieces very small. Place the mixture in a glass bowl and allow it to cool completely in the refrigerator.
Next, warm the milk and lemon peel together in a 4qt pan, just until warmed, do not boil. Let the mixture rest off the heat for 10 minutes to infuse the flavor of the lemon into the milk.
Whisk together the egg, egg yolks, sugar, cornstarch, and salt together until pale and fluffy, about 5 minutes.
Remove the lemon peel from the hot milk, and slowly whisk the warm milk, a couple of tablespoons at a time, stirring constantly, into the egg mixture. When all of the milk is in the egg mixture, put it all back in the saucepan and cook over medium-low heat, stirring constantly until the mixture thickens about 3-4 minutes. Do not boil the mixture.
Whisk the butter and lemon juice into the now custard, place the custard back into the bowl, and cover with plastic wrap, ensuring the wrap touches the top of the custard, refrigerate until cool.
While the custard and jam are cooling, cut and bake the puff pastry. Cut the thawed sheets of puff pastry in half, then cut each half into 6 rectangles. Place the pieces on a parchment-lined baking sheet, prick each piece with a fork all over, place another piece of parchment on top along with a second baking sheet. This ensures the pastry will rise evenly in the oven and keep a thin profile for stacking later. Bake at 450˚F for 15 minutes, then remove the top pan and parchment and bake an additional 5 minutes until golden brown. Allow cooling completely before stacking.
When the custard and strawberry jam is cool, whip the heavy cream to medium peaks and fold it into the custard, along with half of the strawberry jam, leaving some streaks of jam running through the mixture. Place the mixture in a piping back fitted with a large plain round tip, one big enough piece of strawberry won’t get stuck in.
In a blender blend the remaining strawberry jam and the juice of a lemon to create a strawberry lemonade sauce. Add a tablespoon of water if needed to make a nice thin sauce.
To build your Mille-Feuille, place one piece of puff pastry on the bottom, pipe custard/cream mixture onto the pastry in dollops, top with the second piece of pastry, repeat this process and finish with dollops on top of the last piece of pastry. Drizzle more sauce over the top and garnish with strawberry slices, dust with powdered sugar if desired.