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 just 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 disk. 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 to 1/8-inch thickness and place in the prepared pan, trimming the edges, and bake until light golden brown, approximately 20 minutes. Let cool completely. Pour the butterscotch filling into the pie crust that is still inside the tart pan and refrigerate for at least 3 hours. Slice the bananas, making concentric circles and top with sugar. Torch bananas briefly until golden and all the sugar is caramelized.
In a bowl, whisk the cornstarch into 1/2 cup of the milk. Let rest for 1 minute, then whisk again. Whisk in the eggs, yolks, and vanilla. In a saucepan, over medium heat, bring the brown sugar, butter, bourbon, and the remaining 2 1/2 cups milk just to the simmering point (it will start to wiggle in the pot). While constantly whisking, slowly drizzle the hot liquid into the egg mixture. Return the mixture to the saucepan and, while constantly whisking and scraping the bottom of the pot, cook until tiny bubbles boil up for 10 seconds. Remove from the heat and strain.