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Preheat the oven to 350°. In a medium saucepan, cook the carrots in the water over moderate heat until tender, about 5 minutes. Drain and pat dry.
In a medium bowl, whisk the flour with the baking powder and salt. Using an electric mixer, in another bowl, beat the shortening with the sugar at medium speed until light and fluffy. Beat in the egg, then beat in the carrots, lemon zest and vanilla. Beat in the dry ingredients at low speed just until combined.
Spoon 2-tablespoon-size mounds of batter onto 2 ungreased cookie sheets and bake for about 25 minutes on the top and middle racks of the oven, or until golden and risen; shift the sheets from top to bottom and front to back halfway through baking. Transfer the cookies to a wire rack to cool slightly.
The cooled cookies can be stored between sheets of wax paper in an airtight container for up to 3 days.