adapted from Gourmet Magazine
FOR BLUEBERRY COMPOTE
2 cups blueberries (10 ounces)
1/4 cup sugar
2 (3- by 21/2-inch) strips lemon zest
1 tablespoon fresh lemon juice
2 teaspoons cornstarch
FOR SANDWICH LAYERS
1 cup all-purpose flour
1/2 teaspoon baking powder
1/4 teaspoon baking soda
1/4 teaspoon salt
1 stick unsalted butter, softened
3/4 cup packed light brown sugar
1 large egg
1/2 teaspoon pure vanilla extract
FOR LEMON ICE CREAM
2 pints premium vanilla ice cream, or one batch of homemade vanilla ice cream
1 tablespoon grated lemon zest
2 tablespoons fresh lemon juice
MAKE BLUEBERRY COMPOTE:
Cook blueberries, sugar, and zest in a 12-inch heavy skillet over medium-high heat, thoroughly crushing blueberries with a potato masher, until juices are released and sugar has dissolved, about 2 minutes.
Stir together lemon juice and cornstarch, then stir into blueberry mixture. Boil, stirring, 1 minute (mixture will thicken).
Transfer blueberry compote to a bowl and chill until cold, about 1 hour. Discard lemon zest.
MAKE SANDWICH LAYERS WHILE COMPOTE CHILLS:
Preheat oven to 375°F with rack in middle. Butter two 8×8 inch baking pans and line with foil, leaving a 1-inch overhang on each side, then butter foil.
Whisk together flour, baking powder, baking soda, and salt in a small bowl. Beat together butter and brown sugar in another bowl with an electric mixer at medium-high speed until pale and fluffy, about 3 minutes. Beat in egg and vanilla. At low speed, add flour mixture in 2 batches, mixing until just combined.
Divide batter between baking pans and spread into thin, even layers with offset spatula. Bake until golden-brown but still tender, 10 to 12 minutes. Cool completely in pans, about 30 minutes.
MAKE LEMON ICE CREAM:
Let ice cream soften slightly and stir in lemon zest and juice. You can also make homemade vanilla ice cream and add the lemon juice and zest to the bowl towards the end of the churning process.
Spoon all of softened ice cream over 1 sandwich layer (in pan) and spread evenly using clean offset spatula. If ice cream becomes too soft and runny, you can return to freezer for a bit. Dollop tablespoons of blueberry compote all over ice cream, then swirl it gently through ice cream with a spoon, use caution not to cut through cookie. Invert second sandwich layer over ice cream, pressing gently to form an even sandwich. Wrap baking pan in plastic wrap and freeze until firm, at least 2 hours.
Transfer sandwich to a cutting board using overhang. Trim edges if desired, then cut into 8 pieces.