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Sift together the cake flour, bread flour, baking soda, baking powder and salt into a large bowl and set aside.
Cream together the butter and sugars on medium speed until very light, about 5 minutes. Add the eggs one at a time, mixing well after each addition, then add the vanilla. Reduce the mixer speed to low, gradually add the dry ingredients and mix until just combined, 5 to 10 seconds. Using a rubber spatula, fold in the chocolate chips.
Press plastic wrap against the dough and refrigerate for at least 24 hours, up to 72 hours.
When ready to bake, preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a nonstick baking mat.
Scoop 3 1/2-ounces of dough, roll into a rough ball (it should be the size of a large golf ball) and place on the baking sheet. Repeat until you have six mounds of dough on the cookie sheet. Sprinkle lightly with sea salt and bake until golden brown but still soft, 18 to 20 minutes. Transfer the parchment or silicone sheet to a wire rack for 10 minutes, then transfer the cookies onto another cooling rack to cool a bit more, until just warm or at room temperature. Repeat with remaining dough (or keep some of the dough refrigerated for up to 3 days, and bake cookies at a later time). Store leftover cookies in an airtight container at room temperature for up to 3 days.