Let gelato soften at room temperature until pliable, about 10 minutes. Line a 13- x 9-inch pan with aluminum foil. Drop gelato by spoonfuls in prepared pan, and, using an offset spatula, spread in an even layer. Freeze until firm, about 2 hours.
Trace 20 (2-inch) circles 2 inches apart on 2 pieces of parchment paper. Turn over, and place on 2 rimmed baking sheets. Set aside.
Sift together almond meal and powdered sugar until thoroughly combined and fluffy, about 3 times, discarding any larger pieces of almond meal. Set aside.
Beat egg whites with an electric mixer on medium speed until foamy, about 1 minute. Gradually add granulated sugar and cream of tartar, beating until mixture is glossy and holds soft peaks, about 3 minutes.
Add food coloring and almond meal mixture to egg white mixture. Gently fold until batter is smooth and somewhat fluid, about 4 minutes, repeatedly pressing batter against sides of bowl in long, wide strokes.
Transfer batter to a piping bag fitted with a 1/4-inch round tip, and pipe rounds over pre-traced circles on parchment paper in prepared baking sheets. Firmly tap baking sheets on the counter 4 times to release any air bubbles. Set aside until tops of cookies are dry, about 30 minutes.
Preheat oven to 300°F with rack in the bottom third of the oven. Bake cookies, 1 baking sheet at a time, on bottom oven rack of preheated oven until edges of cookies can be gently lifted from parchment, 20 to 22 minutes. Cool completely on pan on a wire rack, about 20 minutes.
Just before serving, cut 10 rounds from frozen gelato, using a 2 1/2-inch cutter, and place each round between 2 cookies.