In a medium bowl, sift together the cake flour, bread flour, baking soda, baking powder, and sea salt. Set aside.
In a stand mixer fitted with the paddle attachment, beat the butter, brown sugar, and granulated sugar on medium-high speed until light and fluffy, about 5 minutes.
Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Reduce the mixer speed to low and gradually add the dry ingredients. Mix only until combined.
Fold in the dark chocolate chips until evenly distributed.
Cover the dough with plastic wrap pressed directly against the surface and refrigerate for 24 to 36 hours. The dough can be refrigerated for up to 72 hours.
When ready to bake, let the dough rest at room temperature for 30 minutes while preheating the oven to 350°F (175°C).
Line a baking sheet with parchment paper or a silicone baking mat.
Arrange five graham cracker squares on the baking sheet. Top each with one piece of chocolate and one marshmallow.
Scoop five 4-ounce portions of dough. Flatten each portion into a large disc and gently wrap it around each graham cracker stack, completely covering the filling.
Sprinkle the tops with flaky sea salt.
Refrigerate the assembled cookies for another 25 minutes to help maintain their thick shape.
Bake the Giant S'mores Cookies for 22–24 minutes, or until golden around the edges while still soft in the center.
Allow the cookies to cool on the baking sheet for 10 minutes before transferring to a wire rack.
Repeat with the remaining dough.