1. In large bowl, mix all-purpose flour, cake flour, and salt. With fingertips, work 1/2 cup butter (1 stick) into flour mixture until coarse crumbs form. Stir in 1 cup ice water until soft dough forms, adding more water if necessary, 1 tablespoon at a time. Wrap dough in plastic wrap; refrigerate 30 minutes. Meanwhile, between 2 sheets of waxed paper, roll 1 1/2 cups butter (3 sticks) into a 6-inch square; wrap and refrigerate.
2. On lightly floured surface, with floured rolling pin, roll dough into a 12-inch square; place butter square diagonally in center. Fold corners of dough over butter so they meet in the center, overlapping slightly. Press with rolling pin to seal. Roll dough into 18" by 12" rectangle. Fold one-third of dough over center, then fold opposite one-third over first to form 6" by 12" rectangle; press seam to seal.
3. Give dough quarter turn. Roll dough into 15" by 8" rectangle; fold into thirds to form 5" by 8" rectangle; wrap in plastic wrap. Refrigerate at least 1 hour. Repeat, rolling dough into 15" by 8" rectangle and folding into thirds, 2 more times. Wrap and refrigerate 2 hours. Repeat rolling, folding, and refrigerating 2 more times, or 6 times in all. After sixth time, wrap well and refrigerate at least 4 hours (or up to 1 week) before using.
4. To make pastry tarts: Cut dough crosswise in half. On lightly floured surface with floured rolling pin, roll dough, half at a time, into rectangle about 20 1/2" by 10 1/2", gently lifting dough occasionally. With sharp knife or pastry wheel, trim edges to make 20" by 10" rectangle. Cut rectangle into eight 5-inch squares; place on large cookie sheet; refrigerate 30 minutes. Repeat with remaining dough half. (Or freezer-wrap and freeze remaining dough up to 6 months; thaw in refrigerator overnight before using.)
5. For each tart, fold a dough square diagonally in half to form a triangle. Starting at folded side, cut 1/2-inch border strip on both sides of triangle, leaving 1/2 inch uncut dough at triangle point so strips remain attached. Unfold triangle. Lift up both loose border strips and slip 1 under the other, gently pulling to matching corners on base; attach points with a drop of water. Refrigerate 30 minutes.
6. Preheat oven to 400 °F. Bake tarts 20 minutes. Reduce oven temperature to 375°F.; brush top of borders with beaten egg. Bake 20 minutes longer or until centers of tarts are lightly browned. Cool tarts on wire rack.