Preheat oven to 350 degrees and place the rack in the middle position. Butter three 13x9 inch jelly roll pans and line with parchment paper; butter the parchment paper.
In the bowl of a stand mixer fitted with a wire whip, beat egg whites at medium-high speed until they begin to hold stiff peaks. With mixer at high speed, gradually add 1/4 cup of sugar, beating until stiff and glossy peaks begin to form. Transfer whipped egg whites to another bowl.
Replace the whip attachment with a paddle attachment. In the now-empty mixing bowl (no need to wash), add the almond paste and 3/4 cup sugar. Beat on medium speed until almond paste is broken up and combined with the sugar, about 4 minutes. Add softened butter and beat until mixture is light and fluffy, about 3-4 minutes stopping once to scrape sides of bowl. Add yolks and almond extract and beat for an additional 2 minutes. Scrape bowl. With mixer running on low, gradually add flour and salt and mix just until no streaks of flour remain.
Stir about a third of egg whites into the almond mixture to lighten. Fold in remaining egg whites until no white streaks remain.
Divide the batter among three bowls. Fold red food coloring into one, green into another, and nothing into the third. Spread the contents of one of the bowls into the three prepared jelly roll pans with an offset spatula. The layer will be thin, about 1/4-inch. Bake 7-10 minutes, until the batter is set. Do not over-bake or allow the cake to brown.
Remove the pans from the oven; loosen the edges of the cakes from the pan with a sharp knife. Allow cakes to cool completely about 15 minutes.
Line a baking sheet with wax or parchment paper. When all layers are baked and cooled, invert the green layer onto the prepared sheet. Remove the baking paper and spread with the apricot preserves. Repeat, this time inverting the white layer onto the green layer, discarding the baking paper, and spreading with raspberry jam. Invert the red layer onto the white layer and discard baking paper.
Cover the assembled layers with plastic wrap and place a baking sheet on top, weighting with a cookbook. Refrigerate at least 8 hours or overnight.
Remove chilled layers from the refrigerator and uncover. In a double boiler over simmering water, melt chocolate coconut oil until glossy and smooth. Pour half of the chocolate over the red layer, spreading to the edges. The chocolate layer will be thin. Let sit at room temperature until chocolate is set, about 15 minutes.
Cover the chocolate with a piece of wax paper and invert onto a baking sheet. Spread the remaining chocolate onto the green layer. Sprinkle evenly with chocolate sprinkles. Let the chocolate set completely, 15-30 minutes
Trim edges with a long serrated knife to make a clean rectangle. With the long end facing you, and your knife parallel to the short edge, cut cake into 3/4-inch strips. Clean knife with hot water (and dry) between each cut to facilitate easy slicing. Cut each strip into 4 cookies, about 2 inches each.
Store the cookies in airtight containers with wax paper or parchment between the layers. Cookies will remain fresh up to 2 weeks at room temperature.