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For the carrot cake doughnuts:
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Preheat oven to 375 F. Lightly grease an 18-count doughnut pan; set aside.
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In a medium sized mixing bowl sift together the flour, ground cinnamon, nutmeg, baking powder, and baking soda; set aside.
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In the bowl of your stand mixer cream together the butter, light brown sugar, and salt. Add egg, vanilla, and yogurt; beat until smooth. Stir in pineapple and carrots. Slowly mix the dry ingredients into the wet, until fully combined.
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Spoon the mixture into a piping bag or a Ziploc bag with the corner cut off. Pipe batter into prepared pan, filling only 1/2 way up the side of the doughnut cavities. Bake for 10 minutes then remove pan from the oven. Cool doughnuts in the pan for 10 minutes, then invert onto a wire rack.
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For the cream cheese frosting:
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In the clean bowl of your stand mixer cream together the butter and cream cheese until smooth. Pour in the pineapple juice and powdered sugar; continue to beat until completely blended and smooth.
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Frost the doughnuts and top with chopped, toasted walnuts.
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For the chopped walnuts:
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Add the chopped walnuts to a frying pan. Cook over medium heat, stirring frequently, until toasted and fragrant, about 3 to 5 minutes.
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Adapted from Life's Simple Measures.