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Combine apricots, water, granulated sugar and 1/4 teaspoon ginger in a small saucepan. Bring to a boil; reduce heat. Cook and stir for 1 minute. Cover; cool completely.
Beat butter in a large bowl with an electric mixer on medium to high speed for 30 seconds. Add brown sugar, the remaining 1/2 teaspoon ginger, baking soda and salt; beat until combined, scraping side of bowl occasionally. Beat in egg white, milk and vanilla. Beat in as much of the whole-wheat flour and all-purpose flour as you can with the mixer. Using a wooden spoon, stir in any remaining flours.
Roll dough between two pieces of wax paper or parchment paper into a 12-by-10-inch rectangle. Remove top sheet of paper. Spread dough with apricot filling, leaving a 1/2-inch border around the edges. Starting at a long side, roll up dough. Pinch edges of dough to seal. Spread pistachios out on a large piece of wax paper or parchment paper. Roll dough log in nuts to evenly coat the outside, pressing in lightly so nuts stick to dough. Wrap dough in wax paper or plastic wrap. Chill for 2 hours.
Preheat oven to 375°F. Coat a baking sheet with cooking spray. Cut dough into 1/4-inch-thick slices. Place slices 1 inch apart on prepared pan. Bake for 8 to 10 minutes or until edges are lightly browned. Transfer cookies to a wire rack; cool.