Whisk together 1 1/3 cups flour, two pinches sugar, and one pinch salt. Then, cut in the butter just until pea-sized chunks remain.
Whisk one egg with the mascarpone and drizzle over the dough. Work until the mixture just comes together, adding a tiny splash of cold water if necessary, and then knead into a smooth disc.
Wrap the dough and refrigerate for 2 hours. Freezing for 30 minutes if you're impatient works just fine, too.
Heat the oven to 375.
Roll out your chilled dough into a rough, 12-inch round, and transfer to a lightly-buttered baking sheet, and hold in the fridge.
Slice your peaches and gently combine with 1/4 cup sugar, the remaining tablespoon of flour, basil (chiffonade--stack the leaves on top of each other, roll them up lengthwise, and slice thin ribbons), and cardamom.
Arrange fruit mixture in the center of your pastry, leaving a 1 1/2-inch border.
Gently fold the dough edge over the fruit, pinching as needed to ensure a sealed border.
Whisk the remaining egg and, using a pastry brush, spread the egg wash over the pastry and finish with a sprinkling of sugar.
Bake 30-35 minutes until thoroughly golden.
While baking, whip your cream, crush your raspberries, and then gently mix the berries into the cream one spoonful at a time. Chill.
Cool the galette and serve warm or at room temperature with a generous scoop of raspberry whipped cream.