Start with the easiest part. In a small bowl, stir together the brown sugar, flour, cinnamon, melted butter, and vanilla. There's no cooking involved here, just mix until it looks like wet sand. Set it aside while you make the dough. It will keep fine at room temperature for the next hour.
In a food processor, pulse the flours, cinnamon, and cold butter cubes together until the mixture looks like coarse crumbs. Add the ice water one tablespoon at a time, pulsing between each addition, until the dough just clumps together. Don't overwork it, or the crust will turn out tough instead of flaky.
Turn the dough out onto a floured surface and roll it into a thin, even rectangle, about 1/8 inch thick. Use a knife or pastry cutter to cut it into evenly sized rectangles. You'll need two matching pieces for every tart, one for the bottom and one for the top layer.
Spoon a small amount of the brown sugar filling onto half of the rectangles, leaving a clean border around the edges. Brush that border with the beaten egg, then place a second piece of dough on top. Press the edges down and crimp them with a fork so the filling can't escape while baking.
Lay the assembled tarts on a parchment-lined baking sheet and poke a few small holes in the top of each one with a fork. Cover and refrigerate for about an hour, or pop them in the freezer for 20 minutes if you're short on time. This step keeps the butter cold and the crust flaky once it hits the oven.
Preheat the oven to 400°F. Bake the chilled tarts for 15 to 18 minutes, until they're lightly golden on top. Let them cool completely on the baking sheet before you touch the frosting, or it will just slide right off.
Whisk the powdered sugar, cinnamon, vanilla, and water together until smooth. Spread a thin layer over each cooled tart, then let them sit uncovered for about two hours so the frosting can set. Sprinkle with cinnamon sugar or a pinch of coarse salt right before it hardens.