1. In a large bowl, combine 2 3/4 cups all-purpose flour with the cake flour, salt, and sugar. With a pastry blender, cut in the butter pieces (1 stick) until the mixture resembles coarse meal, with a few larger clumps remaining. Make a well in the center, and pour in 1 cup cold water and the vinegar, gradually drawing the flour mixture over the water, gathering and combining until mixture comes together to form a dough. If the dough is too dry, add more cold water, 1 tablespoon at a time. Knead gently in the bowl, and form dough into a rough ball. Wrap in plastic and refrigerate until well chilled, at least 40 minutes or up to 2 hours.

2. Sprinkle 2 tablespoons all-purpose flour on a sheet of parchment. Lay remaining 4 sticks of butter on top, side by side; sprinkle with remaining 2 tablespoons flour. Top with more parchment; pound butter with a rolling pin until it is about 1/2 inch thick. Remove top paper, fold butter in half, replace paper; pound butter until it is 1/2 inch thick. Repeat two or three more times until it is pliable. Using bench scraper, shape butter into a 6-inch square; wrap in plastic, and refrigerate until chilled, about 10 minutes.

3. Lightly dust work surface with flour. Roll out dough to a 9-inch round; place butter package in center. Using a paring knife or bench scraper, lightly score dough to outline butter square. Remove butter; set aside. Starting from each side of marked square, gently roll out dough to form four flaps, each 4 to 5 inches long; do not touch square. Return butter to center square; fold flaps over butter. Press with your hands to seal.

4. With the rolling pin, gently pound the dough all over in regular intervals until it is about 1 inch thick; this will soften the dough, making it easier to roll. Working in only one direction (lengthwise), gently roll out the dough to a 20-by-9-inch rectangle, squaring corners with the side of the rolling pin or your hands as you go. Using a dry pastry brush, sweep off excess flour. With short side facing you, fold the rectangle in thirds like a business letter. Turn the dough a quarter-turn clockwise, so the flap opening faces right, like a book. (This completes the first turn.) Roll out the dough again to a 20-by-9-inch rectangle, rolling in the same lengthwise direction; fold dough again into thirds. (This completes the second turn.) Wrap dough in plastic and refrigerate until well chilled, about 1 hour.

5. Repeat rolling, turning, and chilling process for a total of six turns; always start each turn with the opening of the dough to the right. (If at any time, the dough becomes too soft to work with, return it to the refrigerator until firm.) Wrap dough in plastic; refrigerate 3 to 4 hours before using the dough.