Whisk together the flour, baking soda, cinnamon, ginger, salt, pepper, and ½ cup of the cocoa in a medium bowl; set aside.
In a separate bowl, beat the brown sugar, butter, and shortening with an electric mixer on medium-high speed until fluffy, 2 to 3 minutes. Beat in the egg, molasses, and vanilla. Reduce speed to low and gradually add the flour mixture, mixing just until combined (do not overmix).
Heat oven to 350° F with the racks in the upper and lower thirds. Line two baking sheets with parchment. Whisk together the turbinado sugar and remaining tablespoon of cocoa in a small bowl.
Roll 2 tablespoonfuls of the dough into balls; roll in the cocoa sugar. Place on prepared baking sheets, spacing them 2 inches apart. Gently tap the tops of the cookies to flatten slightly (they should be about 1 inch thick).
Bake, in batches, rotating the sheets halfway through, until the edges are just firm and the cookies are cracked on top, 8 to 10 minutes. Cool slightly on the baking sheets, then transfer to a wire rack to cool completely.
Storage suggestion: Keep the cookies in an airtight container at room temperature for up to 5 days.
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