- Make the dough: In a large mixing bowl, whisk together flour, salt, and sugar. Using a bench scraper or pastry blender, begin to roughly cut butter into flour. When largest pieces have been incorporated, use your hands to pinch and flatten butter pieces, continuously coating butter with flour. The mixture will begin to look like crumbs with pea-sized pieces of butter scattered throughout.
- In a measuring cup, combine apple cider vinegar with enough ice water to reach ½ cup. Add liquid, a little at a time, to flour mixture and incorporate by tossing together. (It will become shaggy, but never tacky.) Once dough comes together, form it into a disk, cut in half, and form into two smaller disks. Wrap and refrigerate for at least 1 hour and up to overnight.
- Roll out 1 disk of dough into a circle that extends 1 or 2 inches wider than a pie plate. Transfer into pie plate, leaving an overhang of about 1-inch. Roll out second disk of dough into a large rectangle. Using a ruler and a paring knife, cut out 6 (2-inch wide) strips. Lay strips on a baking sheet lined with parchment paper and refrigerate until ready to assemble.
- Make the filling: Quarter strawberries and add to mixing bowl. (If strawberries are small, halve them.) Sprinkle 3 tablespoons of sugar over strawberries and set aside to macerate for 30 minutes. Strain excess liquid and add shredded apple to the strawberries, then toss with lime zest, juice, and shrub. Combine remaining ½ cup sugar with black pepper, brown sugar, and tapioca flour. Add mixture to strawberries and toss. (It is best to combine tapioca mixture right before you assemble the pie. If mixture is left to sit, it will develop too much liquid.)
- Assemble and bake: Preheat oven to 425 degrees. Whisk together egg yolk and cream for egg wash. Add strawberry mixture to pie shell and top with three vertical strips of lattice. To weave the lattice, fold center vertical strip down and lay one horizontal strip across center of pie. Return vertical strip to its original position. Next, fold two outside vertical strips down and lay another horizontal strip. Return vertical strips to their original position. Repeat this process to add final horizontal strip. Using a pair of scissors or a paring knife, trim lattice edges so that they are flush with edge of pie plate. Take the overhang of the bottom crust and roll up and over the edge to cover the lattice edges. Press the edges to seal, then crimp as desired. Place in freezer for 10-15 minutes.
- Brush egg wash over entire pie. Sprinkle with demerara sugar and place on a parchment-lined baking sheet (to catch drips). Bake on lowest rack until pastry sets and begins to turn golden, 20 to 25 minutes. Lower the heat to 375 degrees and finishing baking pie on center rack until golden and filling is bubbling, 30 to 35 minutes more.