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®, 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.