Let's start with the crust: Put the flour, Parmesan, salt, and pepper in a food processor and pulse until combined. Scatter diced butter and shortening over the top of the mixture and continue pulsing until it becomes coarse. Drizzle with ice cold water and keep pulsing just until the mixture starts to clump together.

Transfer the dough to a work surface. Gather it all together, and knead until dough comes together. Shape the dough into a disc, cover with plastic wrap, and chill for 30 minutes.

Next, the balsamic reduction: Let the vinegar and sugar boil in a small saucepan over medium-high. Cook until reduced to about 1/3 cup then take it off the heat. Add butter, and whisk until well combined.

Now for the fun part, the pesto: Process the basil, the Parmesan, the pine nuts, the pepper, and the garlic in a food processor until finely ground and pasty. While the food processor is running, drizzle 3 tablespoons of the oil through the food chute. Continue processing until well combined.

Preheat the oven to 375°F. Remove the dough from fridge and ditch the plastic wrap. Place the dough on a lightly floured sheet of parchment paper, and roll it into a 14-inch circle. Spread 3 tablespoons of your pesto over the dough, leaving a 2-inch border. Set aside any leftover pesto. Shingle tomato slices and cheese slices in a circular pattern over the pesto. Fold the edges of the dough up and over the filling to partially cover.

Mix together the egg and the water in a small bowl. Brush the egg mixture over the exposed dough and transfer the newly-born galette and parchment to a baking sheet. Bake in the preheated oven until well browned.

Stir together the remaining pesto and 2 tablespoons oil. Drizzle your galette with your desired amounts of pesto and balsamic reduction. You can save any remaining pesto and balsamic for something else if you like. Sprinkle with basil leaves and cut the galette into 6 wedges.