Pulse the oats in a food processor fitted with a metal blade until some of it is fine and some of it is still very coarse. Whisk the oats with the cake flour, all-purpose flour, baking powder, salt and baking soda in a medium bowl.
Cream the brown sugar, butter and granulated sugar in a mixer fitted with a paddle attachment on medium speed until very light, about 5 minutes. Beat in the egg, mixing well. Beat in the vanilla. Reduce the speed to low, add the flour mixture and mix until combined. Fold in the chocolate chips. Wrap the dough in plastic wrap and refrigerate for at least 24 hours and up to 48 hours.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment. Scoop the dough 1 teaspoonful at a time into balls and place 1 inch apart on the baking sheets. Sprinkle lightly with the sea salt. Bake, rotating sheets from back to front and top to bottom halfway through, until golden brown around the edges and paler and soft in the middle, 10 to 12 minutes. Cool 10 minutes on the sheet on a rack, then transfer the cookies to the rack and cool for as long as you can resist. Continue with the remaining dough.
You can make these twice as large if you'd like; just increase the cooking time by about 2 minutes.
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