Put 2 1⁄2 cups flour, 8 tbsp. butter, lard, and half of the salt in a bowl. Using a dough blender, two forks, or your fingers, cut butter and lard into flour mixture, forming pea-size crumbles. Sprinkle about 10 tbsp. ice-cold water over the mixture; stir gently with a fork until dough comes together. (Alternatively, pulse ingredients in a food processor.) Divide dough in half and flatten into disks. Wrap disks in plastic wrap; chill 1 hour before using.
Unwrap one dough disk; transfer to a well-floured surface. Whack it with a rolling pin several times to flatten it out; turn it over and repeat. Roll out dough to form an 11" circle; transfer it to a 9" deep-dish pie pan. (If dough tears and needs mending, dab a little water where it requires patching and "glue" on a piece of dough.) Cover with plastic wrap and refrigerate.
Arrange a baking sheet on lower rack of oven; heat to 425°. Put remaining flour and salt, 3⁄4 cup sugar, berries, lemon juice, zest, and nutmeg into a bowl. Using a wooden spoon, mash gently to make a textured filling; pour into chilled crust; dot with remaining butter; set aside.
Beat egg white and 1 tbsp. water together in a bowl; set aside. Roll out remaining dough disk into an 11" circle; cut into 3⁄4"-wide strips. Arrange strips over pie in a lattice pattern; trim excess dough and crimp edges of pie. Brush crust with egg mixture; sprinkle with remaining sugar. Bake on the middle rack of the oven until crust is just golden, about 15 minutes. Reduce heat to 350°; bake until golden brown and bubbling, about 30 minutes more. Let cool before serving.