Add the liquid ingredients and gently fold into the dry ingredients, trying to moisten everything without making the butter any smaller. Once the ingredients are roughly combined, turn the mixture out onto a work surface and lightly knead together to form a ball of dough. Return the dough to the bowl and cover with plastic wrap. Refrigerate for 2 hours.
Remove the dough from the refrigerator and place on a well-floured work surface. Roll out the dough into a roughly 8-by-16-inch rectangle. (It will look quite shaggy.) Fold in thirds, like a business letter, brushing off excess flour. This is the first turn. Give the dough a quarter-turn and repeat the rolling and folding process 2 more times, giving the dough a total of 3 turns. (It will get increasingly smoother.) Wrap in plastic wrap and refrigerate overnight.
To make the lemon sugar: In a small bowl, rub the 1/3 cup granulated sugar and lemon zest together for a few minutes with your fingers. Set aside.
To make the glaze: Stir in lemon juice, a teaspoon at a time, to the powdered sugar to make a thin glaze.
To make the pastry cream: In a bowl, whisk together the egg yolks, flour and granulated sugar (1/4 cup plus 1 tablespoon).
In a medium saucepan, bring half-and-half just to a boil over medium heat. Whisking constantly, slowly pour the half-and-half over the egg mixture, then pour this back into the saucepan and cook until thickened, continuing to whisk. Whisk in the vanilla. Pour the thickened custard into a clean bowl, cover with plastic wrap and refrigerate until needed.