To prepare crust: Combine all-purpose and whole-wheat flours, 2 tablespoons sugar, baking powder and 1/2 teaspoon salt in a food processor. Pulse 3 times to mix. Add oil, 4 tablespoons butter and cream cheese and pulse until it looks like coarse meal, 10 to 15 times. Add 5 tablespoons water and vinegar; pulse until the dough is starting to come together, about 5 more times; add 1 tablespoon water, if necessary. Turn the dough out onto a clean surface and knead until it comes together. Divide the dough in half and shape into two 5-inch-wide disks. Wrap each in plastic wrap and refrigerate for at least 1 hour.

When ready to assemble & bake: Preheat oven to 425 degrees F.

Roll out one disk of dough on a lightly floured surface into a 12-inch circle. Transfer to a 9-inch pie pan.

To prepare filling: Gently toss apricots, sugar, flour, tapioca, lemon juice, salt and nutmeg in a large bowl. Scrape the filling into the crust. Dot with butter.

To prepare lattice top: Roll out the remaining dough on a lightly floured surface into a 12-inch circle. Cut the dough into about 12 strips (1 inch wide). Lift off every other strip and lay them on top of the pie, leaving about a 1-inch gap between strips. Fold back the first, third and fifth strips of dough all the way to the edge of the pie. Place a shorter strip of dough across the unfolded strips, about 1 inch from the edge. Unfold the folded strips over that crosswise strip. Fold back the second and fourth strips to the crosswise strip. Place another strip crosswise, about 1 inch from the first. Unfold the strips over that second crosswise strip. Continue alternately folding and placing until the top is covered with woven strips. (You may not use them all.)

Trim the crust so it evenly overhangs the edge of the pan by about 1 inch. Tuck the overhang under and press to seal the two crusts together. Crimp the edge with a fork or flute it between your thumb and the side of your index finger. Brush the lattice with egg white mixture and sprinkle with the remaining 2 teaspoons sugar.

Bake the pie for 15 minutes. Reduce oven temperature to 375 degrees F and continue baking until the crust is golden brown, the fruit is tender and the juices are bubbling, about 35 minutes more. Tent with foil if the crust is browning too quickly. Let cool completely on a wire rack before slicing.