Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with a silicone baking mat or parchment paper.
Put butter, white sugar, brown sugar, vanilla, baking soda, and salt in a bowl. Cream together with a mixer until butter clings to sides of bowl and is uniform in color, about 2 minutes. Mix in eggs 1 at a time, waiting until first is incorporated before adding second. Add 2 cups flour; mix on low speed until incorporated. (Dough will be very soft.)
Divide dough roughly in half and transfer one half to a separate bowl. Mix remaining 1/2 cup flour into one half, then stir in semisweet chocolate chips until evenly distributed. Mix cocoa powder into other half, then stir in white chocolate chips until evenly distributed.
Roll a tablespoon of each dough into a ball and tap lightly on counter to create a flat bottom (see assembly shots at left). Press flat sides together to form a larger ball, then flatten into a half-dark, half-light disk. Cut disk in half with a knife, cutting perpendicular to dough seam. Flip one half over and then firmly press halves back together. You should have a disk with 4 alternating quarters. Repeat with remaining dough. Transfer to baking sheet, spacing about 2 inches apart. Sprinkle with flaky sea salt.
Bake until cookie edges start to turn golden brown, 10 to 12 minutes. Cool on sheet 10 minutes. Transfer to a wire rack to cool completely.