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1
Position a rack in the center of the oven and preheat the oven to 375u00b0F. Line two baking sheets with parchment paper.
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2
Sift together the flour, cocoa powder, baking soda, and salt onto a sheet of waxed paper. In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter, shortening, and brown sugar on low speed until just combined. Increase the speed to medium and beat until fluffy and smooth, about 3 minutes. Add the egg and vanilla and beat for another 2 minutes.
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3
Add half of the flour mixture and half of the milk to the batter and beat on low until just incorporated. Scrape down the sides of the bowl. Add the remaining flour mixture and milk and beat until completely combined.
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4
Using a spoon, drop about 1 tablespoon of batter onto one of the prepared baking sheets and repeat, spacing them at least 2 inches apart. Bake one sheet at a time for about 10 minutes each, or until the pies spring back when pressed gently. Remove from the oven and let the cakes cool in the pan for about 5 minutes before transferring them to a rack to cool completely.
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5
In the work bowl of a stand mixer fitted with the paddle attachment, beat the butter, starting on low and increasing to medium speed, until creamy, about 2 minutes. Add the salt, lime juice, and vanilla, and beat until smooth, about 1 minute. Add the confectioners' sugar, 1 cup at a time, and beat on low until incorporated. Increase the speed to medium and beat until light and fluffy, about 3 more minutes. Add more confectioners' sugar as needed to get the consistency you want.
