An homage to the rainbow cookie—and Italian-American classic—these triple-deckers owe their subtle hues to matcha powder and pulverized freeze-dried raspberries, and their unique flavor to ground hazelnuts.
Preheat oven to 350°F. Spread hazelnuts in a single layer on a rimmed baking sheet; bake until golden brown, 6 to 8 minutes. Let cool completely.
Spray three 9 1/2‐by‐12 1/2‐inch rimmed baking sheets (also called quarter‐sheet pans) with cooking spray. Line with parchment; spray parchment.
In a food processor, finely grind cooled hazelnuts. Add confectioners' sugar, 1 egg white, and 1/4 teaspoon salt. Blend until mixture holds together. Transfer to a mixer bowl fitted with a TK attachment. Add 1 cup granulated sugar, and beat on medium speed until combined, about 2 minutes. Add butter and beat until fluffy, about 3 minutes.
Add liqueur to mixer, then egg yolks, one at a time, beating to combine after each addition. Reduce speed to low; beat in flour and remaining 1 teaspoon salt until combined.
In a clean bowl, beat remaining 5 egg whites in mixer fitted with the whisk attachment on medium speed until soft peaks form, gradually adding remaining 1/4 cup granulated sugar. Continue beating until stiff peaks form, 3 to 4 minutes. Whisk into flour mixture in two additions. Divide mixture equally between three bowls (one can be the bowl you mixed in).
Stir ground raspberries into one bowl, matcha into another, and leave third plain. Spread each batter evenly into a separate prepared pan. Bake, rotating pans once, until just set, dry on top, and just beginning to pull away from sides of pan, 18 to 20 minutes. Let cool completely in pans on wire racks.
Turn green cake out of its pan onto a parchment‐lined baking sheet or cutting board. Remove parchment. Spread half of jam evenly over cake, all the way to edges. Turn out white cake onto green cake; remove parchment. Spread remaining jam over white cake. Turn out pink cake onto white; remove parchment. Wrap in plastic and top with a small baking sheet or board and a few cans to weigh down. Refrigerate at least 4 hours and up to 12 hours.
Place chocolate in a heatproof bowl set over a pot of simmering water, or in a microwave. Just before all is melted, remove from heat; stir with a spatula until completely melted. Pour half of chocolate over cake and spread evenly with an offset spatula.
Refrigerate until set, about 10 minutes. Flip over and repeat on second side with remaining chocolate. Refrigerate 10 minutes. Cut into 1 1/4‐inch squares. Refrigerate in a single layer in an airtight container up to 1 week, or store at room temperature up to 2 days.
If you don't have three quarter-sheet pans, work in batches. (If you have none, use a 9-by-13-inch cake pan.)