Special equipment: a 9-inch tart pan with removable bottom
Preheat the oven to 425 degrees F.
Dust a clean work surface with flour. Unroll the pie dough and roll it out just enough to square the rounded edge and make sure it fits a 9-inch tart pan. Lay the dough in the pan, folding any excess dough over itself to create a thick crust. Use a pastry brush to coat the dough with the heavy cream, and then use a fork to dock the bottom all over. Place the tart pan on a baking sheet and blind bake on the center rack of the oven until golden, 10 minutes. Set aside to cool.
In a large mixing bowl, add the butternut squash, 1 tablespoon of the mixed herbs, 1 tablespoon of the olive oil and 1 teaspoon of the salt. Toss together and set aside.
In a large skillet set over medium-high heat, add the butter and the remaining 1 tablespoon olive oil. Add the cremini mushrooms, oyster mushrooms, garlic, 1 tablespoon of the mixed herbs, the remaining 1 teaspoon salt and the pepper. Saute until golden brown, 8 to 10 minutes. Cool the mushrooms slightly and then add the ricotta and lemon zest. Stir together and set aside to cool completely.
Using a rubber spatula or a spoon, evenly spread the mushroom-ricotta mixture in the tart shell. Lay the butternut squash over the top in an overlapping, shingle pattern. Bake until golden brown and the butternut squash is cooked through, 30 minutes.
Garnish with the remaining 1 tablespoon mixed herbs and the grated and shaved Parmesan to serve.