
Preheat oven to 375º F.
Place tapioca flour, 3 tablespoons arrowroot flour, sugar, cinnamon, and salt in the bowl of a food processor. Pulse a few times to thoroughly combine.
Drop in very cold palm shortening, in about 1 tablespoon drops. Pulse a couple times until shortening is incorporated and resembles coarse crumbs.
Pour in egg and pulse a couple times again until mixture again resembles coarse crumbs. Then, add in water, about 1-2 tablespoons at a time, pulsing after each addition, until it’s beginning to form a dough. It shouldn’t be too sticky.
Lay a sheet of parchment paper on your countertop and dust with arrowroot. Turn dough out onto parchment paper and form into a ball. You may need to add arrowroot to work with it. Pull about 1/3 of the dough off the ball, form a smaller ball out of this dough, wrap in plastic wrap, and chill. The total dough will make about 415 grams, and you’ll want to reserve about 140 grams, leaving about 275 grams.
Dust the larger ball with arrowroot and flatten into a disc. Place the second sheet of parchment paper on top. Roll out into a circle, the size of your pie pan. Lightly sprinkle arrowroot onto the edge of your pie pan. Then gently transfer rolled-out crust to pie pan and press down into pan, fluting edges. If the crust breaks, repair it by pressing the dough back together or adding a bit more dough in. It’s very forgiving.
Prick lightly all over with a fork, then grease a sheet of aluminum foil with coconut oil or more shortening then place, greased-side down, in the crust. Fill with pie weights, dried rice or beans. Bake for 5 minutes then remove weights and foil. Bake another 2 minutes. Remove from oven and let cool completely. Now make your filling.
Mix together sugar, 1/2 cup tapioca flour, and salt in a bowl.
Pour 3/4 of the sugar mixture on the strawberries, add 1 tablespoon extra tapioca flour, and vanilla extract. Stir to combine very well. Set aside for 10 minutes to macerate.
Pour the remaining 1/4 of the sugar mixture on the blueberries and add lemon zest and lemon juice. Stir to combine well. If the mixture seems very dry, smash a few blueberries to allow the mixture to coat the blueberries better.
Add in a bit more sugar to the berries, if necessary, depending on the sweetness of your fruit.
Once the strawberries have macerated (softened and produced juice), sprinkle in the remaining 1 tablespoon of tapioca flour and stir to mix very, very well (see note).
Fold a piece of aluminum foil in half width-wise a couple times to make a divider for the fruit. Put the foil inside the pie crust blocking off a third of the inside of the pie (see photo).
Spoon blueberries into the smaller side. Then, using a slotted spoon, spoon strawberries into larger side, leaving strawberry liquid in bowl. Carefully remove the foil.
Roll out smaller ball of dough and cut out a few strips, about 3/4-1” thick. Slide a kitchen knife under the crust strips (above the parchment paper) and lift up to remove. Place over strawberry section to resemble American flag. Cut out stars with cookie cutter and place over blueberry section. Use your fingers or a kitchen knife to easy away the dough around the star, then slide the knife under the star and lift up. Be very gentle, as this dough can be delicate.
In a small bowl, combine whisked egg with 1 teaspoon water. Using a pastry brush, gently brush the top crust and bottom crust edges with the egg wash. Bake for 25-30 minutes uncovered, or until top crust is golden brown. Cover edges or entire pie (depending on how brown the stars and stripes are) and bake another 5-10 minutes, or until filling is bubbly and crust is golden brown. Remove from oven and let cool at least 2 hours before cutting. Perfect with coconut ice cream!