Special equipment: a 3-quart baking dish
Preheat the oven to 375 degrees F with a rack in the center of the oven and a second rack underneath; place a baking sheet on the second rack (to catch drips).
For the filling: In a large bowl, toss the peaches and blackberries with the granulated sugar, cornstarch and lemon juice. Transfer to a 3-quart baking dish.
For the topping: Whisk together the all-purpose flour, almond flour, baking powder, baking soda and salt in a large bowl. Using your fingers or two knives, cut in the butter and lard until a coarse meal forms. Create a well in the center and pour in the buttermilk. Mix until just combined. The dough will seem quite wet, but that's what you want for the topping.
Drop spoonfuls of biscuit dough onto the filling, brush lightly with milk and sprinkle with the raw sugar.
Bake until the fruit is bubbling and tender, and the biscuits are deep golden-brown and cooked through, 50 to 60 minutes. Cool for at least 15 minutes before serving.
Meanwhile, make the whipped cream: Whip the cream and granulated sugar until soft peaks form. Fold in the almonds.
Serve the cobbler warm or room temperature in bowls; top with dollops of whipped cream.