Melt butter in a small heavy-bottomed saucepan over medium heat. Continue cooking until light brown and nutty-smelling, about 6 minutes. Remove from heat and strain through a fine mesh sieve into a bowl.
Sift together confectioners' sugar, pistachio flour (or pistachios and flour, processed in a food processor until fine), cake flour, and salt in a large bowl. Whisk in egg whites to combine. Whisk in strained butter in a slow, steady stream. Cover bowl with plastic wrap and refrigerate until thickened, about 1 hour and up to 1 week.
Preheat oven to 350 degrees. Generously butter a 9-inch fluted tart pan with a removable bottom.
Pour chilled batter into prepared tart pan. Sprinkle grapes over the top, keeping fruit 1/2 inch in from edge of pan. Sprinkle grapes and batter with demerara sugar.
Place tart pan on a baking sheet and bake until edges of cake are dark golden brown and top is slightly springy to the touch, 1 hour 15 minutes to 1 hour 30 minutes. Transfer cake to a wire rack, remove outer ring of pan, and let cool; serve warm or at room temperature. This cake is best eaten the day it is made, but can be kept at room temperature, loosely covered with plastic wrap, up to 2 days.