Instructions

For the pie dough: Grate 4 tablespoons butter on large holes of box grater. Transfer to freezer. Cut remaining 16 tablespoons butter into ½-inch cubes; set aside. Pulse rye flour, sugar, and salt in food processor until combined, 2 pulses. Add cubed butter and process until homogeneous paste forms, about 2 minutes. Using your hands, carefully break paste into 2-inch chunks and redistribute evenly around processor blade. Add all-purpose flour and pulse until mixture is broken into pieces no larger than 1 inch (most pieces will be much smaller), 4 to 5 pulses. Transfer mixture to medium bowl. Add grated butter and toss until butter pieces are separated and coated with flour.
Sprinkle ¼ cup ice water over mixture. Toss with rubber spatula until mixture is evenly moistened. Sprinkle remaining ¼ cup ice water over mixture and toss to combine. Press dough with spatula until dough sticks together. Use spatula to divide dough into 2 even pieces. Transfer each piece to separate sheet of plastic wrap. Working with 1 piece at a time, draw edges of plastic over dough and press firmly on sides and top to form compact, fissure-free mass. Press 1 piece into 5-inch disk. Press remaining piece into 5-inch square. Wrap separately in plastic and refrigerate dough for at least 2 hours or up to 2 days. Let chilled dough sit on counter to soften slightly, about 10 minutes, before rolling. (Wrapped dough can be frozen for up to 1 month. If frozen, let dough thaw completely on counter before rolling.)
Line rimmed baking sheet with parchment paper. Roll dough square into 10½ by 14-inch rectangle on well-floured counter. Transfer dough to prepared sheet and refrigerate for 10 minutes. Meanwhile, roll dough disk into 12-inch circle on well-floured counter. Loosely roll dough around rolling pin and gently unroll it onto 9-inch pie plate, leaving at least 1-inch overhang around edge. Ease dough into plate by gently lifting edge of dough with your hand while pressing into plate bottom with your other hand. Trim overhang to ½ inch beyond lip of plate. Wrap dough-lined plate loosely in plastic and refrigerate until dough is firm, about 30 minutes.
Transfer chilled dough rectangle, still on baking sheet, to counter. Using pizza wheel, fluted pastry wheel, or paring knife, trim ¼ inch of dough from long sides of rectangle, then cut rectangle lengthwise into eight 1¼-inch-wide strips. Refrigerate strips on sheet until firm, about 30 minutes.
For the filling: Adjust oven rack to middle position and heat oven to 400 degrees. Cut vanilla bean in half lengthwise. Using tip of paring knife, scrape seeds into large bowl. Add sugar, cornstarch, lemon zest, cardamom, and salt and whisk to combine. Stir in apricots and lemon juice and let sit for 15 minutes. Spread apricot mixture into even layer in chilled dough-lined plate.
To make lattice, evenly space 4 dough strips across top of pie, parallel to counter edge. Fold back first and third strips almost completely. Lay 1 strip across pie, perpendicular to counter edge, keeping it snug against folded edges of dough strips. Unfold first and third strips over top of perpendicular strip. Fold back second and fourth strips and add second perpendicular strip. Repeat, alternating between folding back first and third strips and second and fourth strips and laying remaining strips evenly across pie to create lattice pattern. Shift strips as needed so they are evenly spaced over top of pie. Trim excess lattice ends, press edges of bottom crust and lattice strips together, and fold under; folded edge should be flush with edge of plate. Crimp dough evenly around edge of plate using your fingers. Brush dough with egg wash.
Bake pie on aluminum foil–lined rimmed baking sheet until crust is golden, 20 to 25 minutes. Reduce oven temperature to 350 degrees and continue to bake until juices are bubbling and crust is deep golden brown, 35 to 50 minutes longer. Let pie cool on wire rack until filling has set, about 4 hours. Serve.