Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with a silicone baking mat or parchment paper.
Peanut Butter Dough: Put butter, white sugar, and brown sugar in a bowl. Cream together with a mixer until butter clings to sides of bowl and is uniform in color, about 3 minutes. Mix in peanut butter, baking powder, baking soda, and salt until incorporated. Scrape down sides of bowl, then mix in egg yolk until incorporated. Add flour; mix on low speed until a dough forms. Cover with plastic wrap.
Ginger-Molasses Dough: Put white sugar, butter, ginger, baking soda, cinnamon, cloves, and salt in a bowl. Cream together with a mixer until butter clings to sides of bowl and is uniform in color. Scrape down sides of bowl; mix in egg until incorporated. Mix in molasses until incorporated; mix in flour until incorporated. Mix in crystallized ginger until incorporated. Scraping bowl between additions.
Roll a walnut-size ball (1 1/2 tablespoons) of ginger-molasses dough in turbinado sugar. Set on prepared baking sheet; flatten slightly with your palm. Brush top with water. Roll a marble-size ball (2 teaspoons) of peanut butter dough, set atop disk, then use a fork to flatten dough and make a crisscross pattern. Repeat with remaining dough. Bake until edges of top cookies start to turn golden brown, 10 to 12 minutes. Cool on sheet 10 minutes. Transfer to a wire rack to cool completely.
Cookies keep up to 3 days in an airtight container at room temperature, or up to 3 months frozen.
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