2 pounds Italian rainbow cookies, chilled until very firm
4 ounces bittersweet chocolate, chopped
1 tablespoon coconut oil
2 pints vanilla ice cream, softened
1 pint dark chocolate ice cream, softened
Special equipment: a 9-inch springform pan
Cut the cookies to form 1-by-1-by-1/2-inch squares. Line a 9-inch springform pan with plastic wrap. Arrange the cookie squares all the way around the sides of the pan with the color lines running perpendicular to the bottom of the pan. Repeat with an additional ring of cookies above the first ring, lining up the colors as much as possible, so the cookies come 2 inches up the sides of the pan and almost completely cover them.
Lay one cookie square in the middle of the bottom of the pan. Arrange a second cookie square next to it, with the color lines perpendicular to the color lines in the first cookie. Continue arranging cookie squares to create a parquet floor pattern. Fill the bottom of the pan with cookies, cutting some to fill in any odd-shaped areas. Chill until cold, about 30 minutes.
Combine the chocolate and oil in a microwave-safe bowl and microwave in 30-second intervals, stirring between each, until melted, about 2 minutes. Using a pastry brush, coat the cookies on the bottom and sides with a thin layer of chocolate. Chill until set, about 5 minutes.
Spread one of the pints of vanilla ice cream in an even layer over the chocolate-coated bottom cookies. Freeze until hard, about 1 hour. Spread the chocolate ice cream in an even layer over the vanilla. Freeze until hard, about 1 hour. Spread the remaining pint of vanilla ice cream in an even layer over the chocolate ice cream. Cover with the remaining cookie pieces. Freeze until solid, 6 hours to overnight.
When ready to serve, invert the cake onto a plate so the bottom layer of cookies is on the top. Remove the pan and peel off the plastic wrap. Cut with a hot dry knife.
Preheat oven to 350 degrees F (175 degrees C). Grease and flour three 9-inch pans. Sift together the flour, baking powder, and salt. Set aside.
In a large bowl, cream together the shortening and sugar until light and fluffy. Beat in the eggs one at a time, mixing until each egg is incorporated; stir in the vanilla. Beat in the flour mixture alternately with the milk. Divide batter into 3 bowls.
In the first bowl, stir in lemon zest and yellow food coloring; pour into prepared pan. In the second bowl, stir in orange zest and orange food coloring; pour into second prepared pan. In the last bowl, stir in the lime zest and green food coloring; pour into third prepared pan.
Bake in the preheated oven until a toothpick inserted into the center of each cake layer comes out clean, about 30 minutes. Let cool in pan for 5 minutes, then turn out onto a wire rack and cool completely.
Assemble the cake: stack the layers together with the Lemon Filling in between the layers. Frost sides and top with Orange Cream Frosting. Refrigerate until serving.