Make the galette dough:
Combine the flour, sugar, ground ginger, and salt in a food processor and pulse to combine. Scatter the chilled butter pieces around the bowl and pulse in 1-second bursts just until the mixture resembles coarse crumbs. Drizzle the water evenly over the crumbs and process just until the dough is moist but still extremely crumbly, about 5 seconds. Turn the dough onto a work surface and press it into a 6-inch disk. Wrap the disk in plastic and refrigerate until well chilled, at least 1 hour.
Make the topping:
In a small bowl, combine the chopped hazelnuts, brown sugar, and flour with a fork until blended. Drizzle the melted butter evenly over the mixture and toss with the fork until combined.
Fill and bake the galette:
Position an oven rack in the middle of the oven; heat the oven to 425°F. Line a rimmed baking sheet with parchment. Unwrap the galette dough and set it on a lightly floured work surface. Roll out the dough, turning and lightly flouring it and the work surface as necessary, into a round that’s 1/8 to 1/4 inch thick and about 15 inches in diameter. Trim the excess dough to make a 14-inch round. Loosely roll the dough around the rolling pin and transfer it to the lined baking sheet. The dough will hang over the edges of the pan. Cover and refrigerate the dough.
Meanwhile, make the filling. In a large bowl, whisk together the cornstarch, brown sugar, and salt until no lumps remain. Add the peach slices, minced ginger, and lemon juice. Toss until the peaches are well combined with the dry ingredients.
Remove the dough from the fridge. If it isn’t pliable, let it warm up at room temperature for a few minutes. Give the peach filling a toss and pile it in the center of the dough, leaving a 3-inch rim bare. Fold this border over the filling, pleating the dough as you go. Gently press the pleats to seal. Tuck in any peach slices that stick out. Brush the pleated border with the cream or milk and sprinkle the hazelnut topping on the dough border, pressing on the nuts so they stick.
Put the galette in the oven and immediately reduce the heat to 400°F. Bake until the crust is browned on the top and bottom, 45 to 50 minutes. Check the galette after about 30 minutes; if the nuts are darkening too much, cover very loosely with foil.