1. Combine 2 3/4 cups chocolate chips, butter and oil in a microwave-safe bowl. Microwave on high, stirring every 30 seconds, until melted, about 2 minutes. Stir in peppermint extract. Let cool for 10 minutes.
2. Meanwhile, whisk all-purpose flour, whole-wheat flour, baking powder and salt in a small bowl.
3. Combine granulated sugar and eggs in a large bowl or the bowl of a stand mixer fitted with the whisk attachment. Beat on high speed until light in color and the volume has doubled, about 3 minutes. Using a flexible spatula, fold in the cooled chocolate mixture, then the flour mixture. Fold in the remaining 1 cup chocolate chips, being careful not to overmix. Refrigerate for at least 1 hour and up to 4 days.
4. Position racks in upper and lower thirds of oven; preheat to 350°F. Line 2 baking sheets with parchment paper.
5. Place confectioners' sugar in a small bowl. Working in batches, scoop dough into balls, each about 1 1/2 tablespoons, and coat in the sugar. Arrange 12 cookies on each prepared pan. Dust your hands with confectioners' sugar and flatten each ball to a 1/2-inch thick round with your fingers.
6. Bake the cookies, switching the pans from top to bottom and front to back halfway through, until just cooked through, 15 to 20 minutes. Let cool on the pans for 10 minutes. Transfer to a wire rack and let cool completely. Repeat with the remaining dough.