How to Make It
Prepare the Crust: Pulse flour, Parmesan, salt, and pepper in a food processor until combined, 2 to 3 times. Scatter diced butter and shortening over top; pulse until mixture resembles coarse sand, 4 to 5 times. Drizzle with ice water; pulse just until mixture starts to clump together, 3 to 5 times.
Transfer dough to a work surface. Gather dough together, and knead until dough comes together, 3 to 4 times. Shape dough into a disc. Cover with plastic wrap, and chill 30 minutes.
Prepare the Balsamic Reduction: Bring vinegar and sugar to a boil in a small saucepan over medium-high. Cook until reduced to about 1/3 cup, 15 to 20 minutes. Remove from heat. Add butter, and whisk until well combined.
Prepare the Pesto: Process basil, Parmesan, pine nuts, pepper, and garlic in a food processor until finely ground and pasty, about 1 minute. With food processor running, drizzle 3 tablespoons of the oil through food chute. Process until well combined.
Preheat oven to 375°F. Remove dough from refrigerator; remove and discard plastic wrap. Place dough on a lightly floured sheet of parchment paper, and roll dough into a 14-inch circle. Spread 3 tablespoons of the pesto over dough, leaving a 2-inch border. Set aside remaining pesto. Shingle tomato slices and cheese slices in a circular pattern over pesto. Fold edges of dough up and over filling to partially cover.
Whisk together egg and water in a small bowl. Brush egg mixture over exposed dough. Transfer galette and parchment to a baking sheet. Bake in preheated oven until well browned, 45 to 50 minutes.
Stir together remaining pesto and 2 tablespoons oil. Drizzle galette with desired amounts of pesto and balsamic reduction. Reserve remaining pesto and balsamic reduction for another use. Sprinkle with basil leaves. Cut galette into 6 wedges.