1. Coat 2 parchment-lined baking sheets with cooking spray. Whisk together flour, granulated sugar, salt, baking powder, and baking soda in a large bowl. Incorporate butter and cold cream cheese into flour mixture with a pastry cutter or your fingers until mixture resembles coarse meal. Transfer to freezer and chill 30 minutes.

2. Preheat oven to 425 degrees, with racks in upper and lower thirds. Transfer half of flour mixture to another large bowl. Add blueberries to 1 bowl and strawberries to other bowl. Gradually stir 6 tablespoons buttermilk into 1 bowl, gently folding with a rubber spatula until a sticky dough forms. (Do not overmix.) Divide dough into 4 even mounds and place 2 inches apart on 1 baking sheet. Repeat with second mixture, remaining 6 tablespoons buttermilk, and other baking sheet. Bake, rotating halfway through, until shortcakes are golden brown, about 20 minutes. Let cool on baking sheets 5 minutes, then transfer to wire racks to cool completely.

3. Meanwhile, beat together remaining 6 ounces cream cheese and confectioner's sugar with a mixer on medium speed until smooth. Gradually add cream, beating just until mixture becomes smooth and thick, about 1 minute. (Do not overbeat.) Serve alongside shortcakes and berries, with confectioner’s sugar for sprinkling.