Position a rack in the middle of the oven and preheat to 375 degrees F. Lightly grease a 9-inch pie plate, including the lip, with butter.
Gently roll each piece of biscuit dough out to about two times its original size. Stretch and press the biscuit dough, one piece at a time, into the pie plate, until you have covered the bottom and sides, pinching together the seams where pieces meet. Prick the bottom with a fork many times. Bake, until the sides are lightly browned and the bottom forms a crust, 6 to 8 minutes. Remove to a wire rack and let cool. Leave the oven on.
Meanwhile, heat the butter in a large skillet over medium heat and add the sausage. Cook, stirring and breaking the sausage up with a wooden spoon, until browned and cooked through, about 6 minutes. Pour off any excess grease and add the half-and-half to the pan, scraping up any bits that may have stuck to the bottom. Remove from the heat.
Whisk the egg and egg yolk together in a medium bowl and slowly add the sausage mixture in batches, whisking constantly, to incorporate. Stir in the scallions and season with 1/2 teaspoon salt and a generous 1/4 teaspoon black pepper.
Scatter the cheese over the biscuit crust, pressing the cheese over any holes that may have formed during baking. Pour the sausage custard over the cheese and bake until the custard is set, 18 to 20 minutes. Serve hot.
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