Place the flour, sugar, and salt in a food processor bowl fitted with the metal ‘S’ blade. Pulse in 1-second intervals to combine.
Add all of the small cubes of chilled butter to the flour mixture in the processor bowl. Pulse until the butter is mixed with the flour without over-mixing. Bits of butter should still be visible, and the dough should look dry and crumbly.
Add several tablespoons of ice-cold water at a time and continue to pulse. Stop adding water when you can gather the dough by hand and it holds together
Place the mixture on clean work surface to form a ball. Cut the ball in half.
Then, using the palm of your hand, press each half into around disk and wrap individually with plastic wrap. Place in the refrigerator for at least three hours, or overnight for best results.
Remove dough disks from refrigerator and unwrap. Lightly flour the work surface and rolling pin. Place dough onto the floured surface.
Roll out the dough from the center, outward in all directions. Apply even pressure as you roll. Periodically, lift dough edges to make sure it's not sticking to the work surface. Add more flour, if needed. Roll dough until it's less than 1/4-inch thick.
Invert the pie pan over the rolled dough to make sure it's large enough to fit the pan.
Lightly butter the bottom and sides of the pie pan before adding the dough.
To place the dough into the pan, gently fold the rolled dough in half, then in half again. Place the folded point of the dough into the center of the pan and unfold.
With your fingertips, gently press the pie dough onto the bottom and up the sides of the plate.
Brush the beaten egg white onto surface of the bottom crust. Add the apricot filling so it reaches almost to the top, spreading evenly.
Roll the second dough disk as described above. Note: Make sure there will be enough dough overhang to attach the top crust to the bottom crust, and to allow for crimping the edges.
Gently lift the dough for the top and place it over the filling. Using a paring knife of kitchen shears, carefully trim excess dough but allow about 3/4 inch to create the edges.
Fold the edge of the dough underneath itself so that it creates a 1/4-inch border that rests on the edge of the pie pan.
To crimp the edges, you can press the tines of a fork lightly around the edge of the pie shell - or -use both index fingers and your thumb to created a scalloped edge - or- any other method you prefer.