"The dough for these addictive, delicate cookies (known as hua sheng bing in Mandarin) begins in the food processor, but you'll want to finish it by hand to keep it from becoming oily. For extra peanut flavor, use unrefined or semi-refined peanut oil, which smells and tastes nuttier. Cookies keep up to 3 days chilled in an airtight container, layers separated by waxed paper, or frozen up to 3 months."
Position a rack in center of oven and preheat oven to 325 degrees F. Line 2 baking sheets with parchment paper.
Pulse peanuts in a food processor until coarse crumbs form. Add confectioners' sugar; process just until mixture looks like fine crumbs and powder, stopping to scrape the bottom if needed. Do not overprocess.
Stir together 2 cups plus 2 tablespoons flour, salt, and baking powder in a large bowl. Stir in peanut mixture until well combined. Drizzle in oil and melted butter. Mix with hands and knead until a coarse dough, akin to Play-Doh(R), forms, about 5 minutes. It will soften as you knead. If dough becomes oily or feels too soft, chill 15 minutes. If dough feels dry, add oil, 1 tablespoon at a time, until workable.
Roll dough into 1-inch balls; arrange 1 1/2 inches apart on prepared baking sheets. Lightly beat egg with water in a small bowl. Working with 1 sheet at a time and covering remaining sheet until ready to bake, lightly brush cookie tops with egg mixture. Press a whole peanut gently into the top of each ball.
Bake until light golden, 20 to 22 minutes. Transfer to wire racks to cool completely.