For the dough: Combine the flour, sugar, cornmeal and salt in a food processor. Add the butter and pulse to a fine meal. Add the water and pulse just until the dough comes together. Form into a 6-inch disk. Wrap in plastic and chill for 1 hour.
Once chilled, roll the dough into a 10-inch circle. Refrigerate/keep cold until ready to use.
For the cream: Place the pistachios on a baking sheet and warm in an oven at 350 degrees F for 4 to 5 minutes (this will help bring out their bright green color). Set aside and cool.
In a food processor, combine the pistachios and powdered sugar and pulse to a coarse meal. Add the butter, flour, brandy, eggs and egg yolk and blend until fully combined and smooth.
To build the galette: Cut the nectarines off of the pit and slice into 1/2-inch wedges. Toss with the sugar, orange and lemon juices and set aside.
Place the rolled dough onto a parchment-lined baking sheet and spread the pistachio cream in an even layer in the center, leaving a 3-inch border of unfilled crust. Arrange half the nectarines on top of the pistachios, then half of the mulberries. Finish with the remaining nectarines and mulberries. Fold the dough over the fruit and refrigerate (or freeze) until the dough is well chilled.
When ready to bake, preheat the oven to 450 degrees F.
Brush the edges of the galette dough with cream and sprinkle with sugar. Bake until the crust is deep golden brown and the fruit is cooked through, 40 to 55 minutes.