Special equipment: An 8-inch-square baking dish
For the dough: Combine the flour, Parmesan, sugar and salt in a food processor and pulse until mixed. Add about one third of the chilled butter and process until thoroughly combined (the mixture will be slightly yellow). Add the remaining chilled butter and pulse until the mixture resembles a very coarse meal with pea-size bits of butter. Add the parsley and vinegar in a single pulse. Gradually add the ice water through the feed tube pulsing just until evenly combined. Squeeze a handful of the dough together, it should just hold its shape and be a little crumbly. If still very powdery, pulse again, adding up to 2 tablespoons ice water.
Turn the dough out onto a work surface. Pat about two thirds of the dough into a square about 1/2-inch thick. Repeat with the remaining third of dough. Wrap each piece tightly in plastic wrap. Refrigerate until firm, at least 1 hour or up to 2 days.
Adjust the oven rack to the bottom position and preheat the oven to 400 degrees F. Butter an 8-inch-square baking dish.
For the collard filling: Bring a large pot of salted water to a boil. Add the collard greens in batches, stirring between each addition to submerge the greens. Return to a boil and cook until the greens are bright green and just tender, about 5 minutes. Drain and shake off any excess water. Let cool completely. Gently squeeze out only some of the excess liquid from the greens and transfer to a large bowl. Add the sausage, scallions, butter, hot sauce and 3/4 teaspoon salt and toss to combine.
For the corn filling: Whisk together the flour, cornmeal, egg, melted butter, sugar, baking powder, salt and 3/4 cup of the cream in a medium bowl until smooth. Stir in the Cheddar.
To assemble the pie: Let the dough pieces stand at room temperature for a few minutes until slightly softened to make rolling easier. Place the larger piece of dough between two large pieces of floured parchment. Roll into a rough 14-inch square (if the dough gets too soft, pop it in the fridge to firm up, then continue rolling). Roll the smaller piece into a rough 12-inch square.
Transfer the larger piece of dough to the prepared baking dish. Gently press the dough into the bottom and up the sides of the dish, leaving some overhang. Brush the bottom and sides with the remaining 1 tablespoon cream. Add the corn filling and spread evenly. Pile the collard filling on top and spread evenly. Put the remaining dough over the filling, lining up the edges with the bottom piece of dough as best you can. Pinch the edges together, trim any excess, then fold and crimp the edges to form a tight seal. Brush the top and edges of the crust with the egg and cut decorative slits in the top of the pie.
Bake until the crust is deep golden brown and the filling bubbles out a little through the slits, 50 to 60 minutes. (If the crust browns too quickly, cover the pie loosely with foil.) Let cool at least 1 hour before serving.
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