Place the flour, 2½ tablespoons of the granulated sugar, and salt in a large mixing bowl. Whisk to combine.
Add the cold butter. Using a pastry blender or 2 forks, cut the butter into the flour mixture until crumbly, with very small pieces of butter distributed throughout.
Pour the very cold water into the bowl and stir just until the dough begins to come together.
Lightly flour your work surface and turn out the dough. Shape it into a ball, then flatten into a round disc.
Wrap the dough in plastic wrap (cling film) and refrigerate for at least 30 minutes.
Meanwhile, toss the sliced pears with the remaining 3 tablespoons of granulated sugar, vanilla, cinnamon, and cardamom. Refrigerate while the dough rests.
Preheat the oven to 400°F/200°C/Gas 6.
Place a sheet of parchment paper (baking paper) on your work surface. Unwrap the chilled dough, place it onto the paper, and roll into a 10-12-inch (25-30 cm) circle.
Arrange the pear filling in the center of the dough, leaving a 3-inch (7.5 cm) border around the edge.
Fold the dough over the pears, working around the galette and overlapping the dough slightly as needed. Gently press together any cracks.
Whisk the egg with 1 tablespoon water or milk. Brush the exposed crust with the egg wash and sprinkle with turbinado sugar.
Transfer the galette and parchment paper to a baking sheet. Bake for 35-40 minutes, until the crust is deep golden brown.
Allow to cool for 5-10 minutes. Dust lightly with powdered sugar before serving. Serve warm and enjoy!