These whoopie pies merge the best of carrot cake and cheesecake: tender, sweet cake and a creamy, tangy filling. They're best eaten chilled; make ahead of time and refrigerate.
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Add the butter and sugars to the bowl of a stand mixer fitted with a paddle attachment (or a large bowl if using a hand mixer) and beat on medium-high speed until light and fluffy, about 4 minutes. Add the eggs, one at a time, beating well after each addition and scraping down the side of the bowl as needed. (It's OK if the mixture looks curdled.) Beat in the carrots and vanilla. Adjust the speed to medium-low, add half the flour mixture, the milk and the remaining flour mixture and beat until just incorporated.
Spray the prepared baking sheets with cooking spray. Drop mounds of batter (2 heaping tablespoonfuls each) about 2 inches apart on the trays. With damp fingers, smooth and shape the batter into neat 2 1/2-inch mounds. Chill for 30 minutes.
Meanwhile, position an oven rack in the center of the oven and preheat to 375 degrees F.
Bake the chilled cookies, one sheet at a time until lightly golden, 10 to 12 minutes, rotating the pan halfway through (cookies will spread and flatten in the first half of baking time, then rise in the second half). Keep the cookies chilled until ready to bake, and make sure the oven is at 375 before baking the second batch. Let the cookies cool on the pans for 5 minutes, then transfer to a rack to cool completely.
For the filling: Meanwhile, combine the cream cheese, sour cream, granulated sugar, vanilla and lemon zest in the clean mixer bowl and beat on medium-high until smooth. Refrigerate until ready to use.
Dollop about 2 tablespoons of the filling on the flat side of 8 of the cookies, spread it almost to the edges and sandwich with the remaining cookies, pressing slightly. Roll the edges in chopped pecans and refrigerate until chilled, at least 1 hour. Serve chilled.