In a large bowl, stir the flour and sugar together. Using a pastry blender or your fingertips, work the butter into the mixture until it resembles coarse meal. Sprinkle on 3 to 4 tablespoons of the eggy water, and mix it in with your fingers until the dough comes together into a ball. Working on a lightly floured surface, take egg-sized pieces of the dough and smear them away from you with the heel of your hand into 6-inch streaks. Scrape up all the streaks of dough and pile them on top of one another to form a disc. Wrap in plastic and refrigerate for at least 2 hours or overnight.

Set a rack in the middle of the oven and preheat to 400 degrees F. Lightly butter a 9 1/2-inch tart pan with a removable bottom.

On a lightly floured surface, roll the dough into an 11 1/2-inch circle. Fit the dough into the prepared pan. Trim the excess dough to a 1-inch overhang and double the dough over itself to reinforce the sides (save the trimmings for any emergency patchwork). Refrigerate for 20 minutes.

Line the tart shell with aluminum foil and weight down with dried beans or pie weights. Bake for 20 minutes. Remove the foil and bake for 5 minutes longer or until lightly browned. Patch any visible holes in the shell with the reserved dough scraps. Brush the shell with Egg Wash and return to the oven for 5 more minutes, until the wash is shiny and no longer runny. Transfer to a rack to cool and lower the oven temperature to 325 degrees F.

Spread the cashews in the partially baked shell and cover with the Cashew Filling, making sure the nuts are covered. Bake for 30 minutes or until little bubbles appear around the edges of the tart and the center looks barely set. Transfer to a rack to cool.

With a fork, mix together the water, flour, and egg yolk in a small bowl to make an egg wash. Cover and refrigerate.

In a large bowl, lightly beat the sugar and eggs together. Add the corn syrup, rum, and vanilla and beat just to a blend. Mix in the melted butter.