Preheat oven to 350°F. Line two baking sheets with parchment paper and set aside.
Beat the room-temperature butter until smooth.
Add the packed brown sugar, the 2 tablespoons sugar, the molasses, and the vanilla; beat until well blended.
Add the room-temperature egg and mix until fully combined, scraping the bowl sides as needed.
Sift together the flour, the Dutch-process cocoa powder (1/2 cup plus 1 tablespoon), and the salt. Add the sifted dry ingredients to the butter mixture and mix just until incorporated—do not overmix.
Stir in the bittersweet chocolate pieces and the semisweet chocolate pieces until evenly distributed.
Let the dough rest at room temperature for 5–10 minutes, or cover and chill in the refrigerator for 1 hour to help prevent flat cookies.
Drop rounded teaspoons of dough onto the prepared baking sheets, spacing them about 2 inches apart.
Bake at 350°F for about 9 minutes, or until the cookies are just firm at the edges and slightly soft in the centers.
Cool the cookies on the baking sheets for about 5 minutes, then transfer them to a wire rack to finish cooling.