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To make the cookies: Preheat the oven to 375°F. Lightly grease (or line with parchment) two baking sheets.
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In a large bowl, stir together the cake mix, butter, and eggs. The mixture will look dry at first but will eventually become a stiff dough.
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Drop the dough by tablespoonfuls onto the prepared pans, leaving about 2” between them. (A tablespoon cookie scoop is a helpful tool here.)
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Bake the cookies until the edges are set and the bottoms are lightly golden, 10 to 12 minutes; they’ll hardly have any color on top and their centers will look soft and slightly underdone.
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Remove the cookies from the oven and allow them to cool for 5 minutes on the pan before transferring them to a rack to cool completely.
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To make the frosting: In a large bowl, beat together the butter and salt until fluffy.
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Add about half the confectioners' sugar (227g) and beat slowly until well blended.
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Add the vanilla and half the milk or cream (28g) and beat until fluffy.
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Beat in the remaining sugar, then the remaining milk. Once both are added, beat until the frosting is light and fluffy.
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Spread about 1 1/2 tablespoons of frosting onto each cooled cookie.
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Storage information: Store frosted cookies in a tightly sealed container for five days at room temperature. For longer storage, freeze the unfrosted cookies and frosting separately; frost just before serving.