In a food processor, whirl powdered sugar and peanuts until nuts are very finely ground.
In a 3- to 4-quart pan over medium heat, bring 2 to 3 inches of water to a boil; adjust heat to maintain a very low simmer. In a large bowl or the bowl of a standing mixer, whisk sugar into egg whites. Set bowl over simmering water in pan (bottom of bowl should not touch water) and stir constantly until sugar is dissolved and mixture feels warm to the touch. Remove bowl from water and, with an electric mixer on high speed (fitted with the whisk attachment if using standing mixer), whip egg white mixture until thick, stiff peaks form. Gently fold in powdered sugar mixture.
Spoon mixture into a pastry bag fitted with a 1/2-inch plain tip and, with the tip almost touching the parchment, pipe into flat 2-inch circles about 1/8 inch thick, 1 inch apart, on two cooking parchment-lined 12- by 15-inch baking sheets (see notes).
Bake cookies in a 300° oven until tops are shiny and dry and edges are cracked, about 15 minutes; if baking two pans at one time, switch pan positions halfway through baking. Let cookies cool completely on sheets (about 1 1/4 hours), then remove by gently lifting them up and peeling the parchment away from the bottoms.
Spread the flat side of each of half the cookies with about 1 teaspoon chocolate ganache. Top each with a second cookie, flat side toward filling.
In a heatproof bowl set over a pan of barely simmering water (bottom of bowl should not touch water), occasionally stir 6 ounces chopped bittersweet or semisweet chocolate, 2 tablespoons whipping cream, 1 tablespoon corn syrup, and 1 tablespoon butter until chocolate is melted and mixture is smooth. Remove from heat and let cool until thick but not firm, about 15 minutes. Makes about 1 1/4 cups.