Put the white chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Whisk the flour, cocoa powder, baking powder and salt in a medium bowl; set aside.
Whisk the eggs, vegetable oil and food coloring in a large bowl; whisk in the granulated sugar and melted white chocolate until smooth. Stir the flour mixture into the egg mixture with a wooden spoon until combined. Transfer the dough to a piece of plastic wrap and pat into a rectangle; tightly wrap and refrigerate until firm, at least 2 hours or overnight.
Position racks in the upper and lower thirds of the oven and preheat to 300 degrees F. Line 2 baking sheets with parchment paper. Put the confectioners' sugar in a bowl. Roll tablespoonfuls of dough into balls; toss the balls in the confectioners' sugar to coat. Slightly flatten the balls, then dip again in the confectioners' sugar. Arrange the cookies 2 inches apart on the prepared baking sheets. Bake, switching the pans halfway through, until the cookies are puffed and firm, 16 to 18 minutes. Let cool completely on the baking sheets.