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Preheat the oven to 350ºF. Dust a work surface with the flour.
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Roll out 1 dough into at least an 11 by 15-inch rectangle 1/8 inch thick. Transfer to a 9 by 13-inch rimmed baking sheet and trim the edges to a 1/4-inch overhang. Refrigerate while making the filling.
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In a large bowl, mix the strawberries, raspberries, cornstarch, granulated sugar, lime zest and juice, vanilla, and salt. Tip the filling into the chilled crust. Refrigerate while making the lattice top.
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Roll out the 1 remaining dough into a 10 by 14-inch rectangle 1/8 inch thick. Using a knife or a pizza cutter, cut the dough lengthwise into 1-inch strips. Lay half of the strips diagonally over the pie, spacing them about 1 inch apart and cutting the strips in half toward the edges. Fold every other strip back on itself. Lay the remaining strips diagonally in the opposite direction. Alternate the strips in a basket-weave (over-and-under) pattern. Trim the strip ends to a 1/4-inch overhang. Tuck the lattice edges between the bottom crust and the baking sheet and firmly crimp with your fingers to seal.
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If the pie has been out for a while and the dough is starting to soften, refrigerate the pie for 10 minutes.
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Brush the lattice crust with the beaten egg and sprinkle with coarse sugar. Line a large baking sheet with parchment paper or aluminum foil. Place the pie on the prepared baking sheet and bake until the crust is browned and crisp, 50 to 60 minutes. Let cool for at least 45 minutes, slice, and serve.