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