Equipment: 2 (8-inch) square baking pans (2 inches deep); a small offset spatula
Cook blueberries, sugar, and zest in a heavy saucepan over medium-high heat, thoroughly crushing blueberries with a potato masher, until juices are released and sugar has dissolved, about 5 minutes.
Stir together lemon juice and cornstarch, then add to the blueberry mixture. Bring to a boil, and continue stirring for about 1-2 minutes (mixture will thicken).
Transfer blueberry compote to a bowl or jar and chill until cold, about 1 hour. Discard lemon zest.
Cookie Sandwiche Layers (make while compote chills)
Preheat oven to 375°F with rack in middle. Butter baking pans and line with parchment, leaving a 1-inch overhang on each side, then butter parchment.
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, 12-14 minutes. Cool completely in pans, about 30 minutes.
Lemon Ice Cream:
Let ice cream sit at room temp for about 5-10 minutes or transfer ice cream to a microwave-safe bowl and microwave at 30 percent power in 10-second intervals, stirring, until softened, about 50 seconds total (see cooks’ note, below).
Stir in lemon zest and juice. Dollop tablespoons of blueberry compote all over ice cream, then swirl it gently through ice cream with a spoon. Return the ice cream to the freezer.
Spoon all of ice cream over 1 sandwich layer (in pan) and spread evenly using clean offset spatula. 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.
If you don’t have a microwave, ice cream can be softened at room temperature. Ice cream sandwich (before cutting) can be made 1 week ahead and frozen in pan, wrapped in plastic wrap and then in foil. !from Gourmet Magazine August 2009