The day before you intend to bake the pie, begin the filling:  Wash and dry the lemons very well (remove any wax coating if the lemons are waxed.) Using a very sharp knife or a meat slice or mandoline, slice the lemons as thinly as possible, removing any seeds as you come across them. Place the lemon slices in a nonreactive bowl and toss with the sugar and salt. Cover the bowl with plastic wrap and allow to sit out overnight before proceeding with the recipe.

Prepare the pastry: Combine the flour, salt, and sugar in a large bowl. Add the butter and shortening and cut into the flour using a pastry cutter until pea-sized pieces remain. Drizzle in the water and vodka little by little, tossing the dry ingredients so that they are evenly coated with the water mixture. Using your hands, gently work the dough until it just comes together, taking care not to overwork it. Gather the dough into two balls roughly the same size, and wrap each ball in plastic wrap, flattening the dough in the process to form a disk. Refrigerate the dough for at least 30 minutes and up to overnight.

When you are ready to bake the pie, preheat the oven to 425ºF.

Remove the dough from the refrigerator and allow to come to room temperature briefly, then roll each portion of dough out to 1/8-inch thickness. Place one portion of dough into a deep-dish pie pan, easing the sides and bottom down into the pan. Trim the dough so that it overhangs by about 1-inch. Place the second rolled portion of dough onto a baking sheet and then place it and the pie plate in the refrigerator while you finish preparing the filling. 

Add the eggs, egg yolk, flour and melted butter to the macerated lemon slices and mix thoroughly to combine.

Remove the pie pan and top portion of dough from the refrigerator. Transfer the filling to the pie pan. Gently place the top portion of dough on top of the filled pie, trimming edges so that they match with the edges of the bottom portion of dough. Bring the edges together and fold them under to form a top edge. Crimp or press the edge decoratively to seal the edges. Using a sharp paring knife, cut slits in the top of the pie to allow steam to escape as it bakes.

Brush the top of the pie with the tablespoon of cream, then sprinkle the top liberally with the coarse sugar.

Transfer the pie to the oven and bake for 30 minutes. Reduce the heat to 325ºF and continue to bake the pie until deep golden brown, about 30 minutes longer. (Tent the top of the pie if it gets too dark before the time is up.)

Remove the pie from the oven and transfer to a wire rack to cool. Cool completely before serving.