In a food processor, pulse together flour, ground hazelnut, sugar, baking powder, and salt. Pulse in butter just until mixture looks like small pebbles. Drizzle in heavy cream, and pulse just to combine.
Transfer dough to a lightly floured surface and pat dough together, incorporating any stray or dry pieces. Put dough into a large bowl and refrigerate for 30 minutes.
In a large bowl, toss together plum slices, blackberries, sugar, cornstarch, and vanilla bean. Let sit for 30 minutes to macerate.
Preheat oven to 400 degrees F.
Transfer the fruit mixture to a large baking dish. It should fill the dish halfway to three-quarters full, leaving a good inch of space for the biscuit topping.
Use a large ice cream scoop and scoop biscuit topping onto fruit filling leaving some spaces in between the biscuit.
Brush biscuit topping with heavy cream. Sprinkle demerara sugar and chopped hazelnuts over the top.
Place the cobbler on a rimmed baking sheet to catch drips, or place aluminum foil beneath it in the oven.
Place baking sheet on middle rack and bake at 400 degrees F until the topping is turning golden brown around the edges and the fruit filling is bubbling.
Remove cobbler from oven and let it cook for at least 15 minutes before serving.