Preheat oven to 375°. In the bowl of a food processor, combine 3 cups flour, 3 tbsp. sugar, baking powder, and salt; whirl until blended. Add 1/2 cup butter and pulse until fine crumbs form. Add cream and pulse just until dough comes together when you squeeze a clump.
Turn dough onto a floured surface and press to flatten slightly. Sprinkle chocolate chips over the top, then knead until incorporated, 6 to 8 turns. Pat dough out to 1 1/4 in. thickness. With a 2 3/4-in. round cutter, cut out shortcakes; gather dough and pat out again as needed to cut all 6 cakes, dusting surface with more flour as needed to prevent sticking. Set shortcakes slightly apart on a baking sheet, brush tops with melted butter, and sprinkle generously with sugar.
Bake shortcakes until golden brown, 22 to 25 minutes. Let cool on baking sheet at least 15 minutes.
Meanwhile, in a bowl, gently mix 3 tbsp. sugar, berries, and mint.
With a serrated knife, carefully slice shortcakes in half horizontally. Set bottoms on plates and top with berry mixture, dark chocolate sauce, and whipped cream. Set tops in place.