Preheat the oven to 350°F and butter a 13- by 9-inch baking pan. Line the bottom with wax paper, leaving a 2-inch overhang, and butter the paper.
Beat the egg whites in a mixer fitted with a whisk attachment at medium-high speed until stiff peaks form. Gradually add 1/4 cup of sugar while beating at high speed until glossy peaks form. Transfer to another bowl.
Switch to a paddle attachment and beat together the almond paste and remaining 3/4 cup of sugar until well blended, about 3 minutes. Add the butter and beat until pale and fluffy, about 3 minutes.
Add the yolks and almond extract and beat until combined well, about 2 minutes. Reduce speed to low, then add the flour and salt, mixing until just combined.
Fold half of the egg white mixture into the almond mixture to lighten it, then fold in the remaining whites gently but thoroughly. Divide the batter among 3 bowls.
Stir red food coloring into one bowl and green food coloring into another, leaving the third batch plain. Chill the green batter, covered.
Pour the red batter into the prepared pan and spread evenly with an offset spatula (layer will be about 1/4 inch thick). Bake the red layer for 8 to 10 minutes, until just set. Using the paper overhang, transfer the layer to a rack to cool, about 15 minutes.
Clean the pan, then line with parchment or wax paper and butter the paper. Bake the white layer in the prepared pan until just set. While the white layer bakes, bring the green batter to room temperature. Transfer the white layer to a rack.
Prepare the pan as above, then bake the green layer in the same manner as before. Transfer the green layer to a rack to cool.
When all layers are cool, invert the green layer onto a parchment or wax-paper-lined baking sheet. Discard the paper and spread with half of the apricot preserves. Invert the white layer on top of the green layer, discarding the paper. Spread with the remaining preserves. Invert the red layer on top of the white layer and discard the paper.
Cover with plastic wrap and weight with a large baking pan. Chill for at least 8 hours. Remove the weight and plastic wrap.
Bring the layers to room temperature. Melt the chocolate in a double boiler or a metal bowl set over a saucepan of barely simmering water, stirring until smooth. Remove from heat.
Trim the edges of the assembled layers with a long serrated knife. Quickly spread half of the chocolate in a thin layer on top of the cake. Chill, uncovered, until the chocolate is firm, about 15 minutes.
Cover with another sheet of wax paper and place another baking sheet on top, then invert the cake onto the sheet and remove the paper. Quickly spread with the remaining chocolate. Chill until firm, about 30 minutes.
Cut lengthwise into 4 strips and then cut strips crosswise into 3/4-inch-wide cookies. Cookies keep layered between sheets of wax paper or parchment in an airtight container at room temperature for 2 weeks or even longer in the freezer.