Strawberry-Basil Shortcakes

Preheat oven to 400°. Whisk sugar, baking powder, salt, and 2 cups flour in a large bowl. Using your fingers or a pastry cutter, work in butter until the texture of coarse meal with a few pea-size pieces of butter remaining. Add cream and mix until dough just comes together (it will be sticky).
Turn out dough onto a lightly floured surface and pat into a 12x4" rectangle about 3/4" thick. Cut out rounds with a 2 1/2" biscuit cutter, re-rolling scraps as needed to make 8 rounds. Whisk egg with 1 tablespoon water in a small bowl. Transfer rounds to a parchmentlined baking sheet and brush tops with egg wash. Bake until tops are golden brown and shortcakes are cooked through, 15–20 minutes.
DO AHEAD: Shortcakes can be made 2 days ahead. Store tightly wrapped at room temperature.
Coarsely chop 2 cups strawberries (use any bruised or less perfect ones) and cook with 2 tablespoons sugar in a medium saucepan over medium heat, stirring occasionally, until berries are softened and mixture is slightly thickened, about 5 minutes. Let berry compote cool.
Meanwhile, toss basil, 1 tablespoon sugar, and remaining 1 cup strawberries in a medium bowl and let sit until fruit begins to release juices, 10–15 minutes. Discard basil.
Using an electric mixer, beat cream, crème fraîche, and remaining 1 tablespoon sugar to soft peaks, about 4 minutes.
Split shortcakes and fill with berry compote, whipped cream mixture, and macerated strawberries.