Cream butter and 1/4 cup of the sugar in the bowl of an electric mixer; add 2 of the eggs and beat until smooth. Reduce mixer speed and add flour a heaping half cup at a time to form stiff but sticky dough. Turn onto a floured surface and form into a ball. Divide dough into 2/3 and 1/3 portions; wrap each in plastic and press to flatten into a round. Refrigerate at least 1 hour and up to 3 days. When ready to bake, adjust oven rack to lower-middle position and heat oven to 350 degrees. Toss apples with remaining 2/3 cup of sugar and the cloves. Roll the larger dough round on a floured surface to a 13-inch circle; fit into a 10-inch tart pan or a 9 1/2-inch deep-dish pie plate. Turn apples into pie plate. Roll remaining portion of dough on a floured surface into a 10-inch circle. Lay pastry over apples and pinch with bottom piece of dough to seal; flute edges. Brush with an egg wash, and decoratively slit the dough top with a paring knife. Bake until pie is golden brown, 45 to 60 minutes. Remove to a wire rack to cool. Cut into slices and serve.
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