Place 3 sticks unsalted butter in the bowl of a stand mixer (or large bowl if using an electric hand mixer). Let sit at room temperature until softened, about 1 hour.
Arrange 2 racks to divide the oven into thirds and heat the oven to 350°F. Coat 3 (9x13-inch) baking pans or quarter sheet pans with cooking spray, then line each one with parchment paper, leaving a 2-inch overhang on two opposite sides like a sling. (See note below.)
Crumble 7 ounces almond paste into the bowl of butter. Add 1 1/4 cups granulated sugar and 1/2 teaspoon kosher salt. Mix with the paddle attachment on medium speed until smooth, about 5 minutes. Add 4 large eggs and 2 teaspoons almond extract and mix on medium speed until smooth, about 2 minutes. Add 2 cups all-purpose flour and mix on low speed, scraping down the sides of the bowl as needed, until just combined.
Divide the batter equally into 3 separate medium bowls (about 14 ounces each). Add about 5 very small drops red gel food coloring to one bowl of batter and stir until well-combined, adding more coloring as needed until the batter is a deep salmon pink. The cakes will be much brighter after baking. Repeat adding 5 very small drops green gel food coloring into a second bowl of batter until the batter is a bright green. Leave the third bowl of batter undyed.
Transfer each bowl of batter into a separate baking sheet and evenly spread with a small offset spatula or rubber spatula. Don’t worry if the top isn't perfectly smooth — they will even out in the oven.
Bake, rotating halfway through, until the tops look dry but not browned, 10 to 14 minutes. Transfer pans to wire racks to cool completely, about 45 minutes. If using 1 pan, let the cake cool in pan for 5 minutes before lifting the cake out to cool on a wire rack. Alternatively, place them in the refrigerator or freezer to cool faster.
Transfer the green cake to a large cutting board. Top with 1/2 cup apricot jam and spread into a thin layer. Remove the parchment from the undyed layer and place on top of the green layer. Top with the remaining 1/2 cup apricot jam. Remove the parchment from the pink layer and place on top of the undyed layer.
Cover the cake with another piece of parchment paper. Place a baking sheet on top of the cake and weigh it down with a few cans. Refrigerate for at least 2 hours or up to overnight.
Place 1 1/2 cups semi-sweet chocolate chips in a medium microwave-safe bowl. Microwave in 30-second intervals, stirring in between each interval, until nearly melted, about 90 seconds total. Stir until completely smooth.
Remove the cans, baking sheet, and top layer of parchment paper from the cake. Using a sharp knife, trim the edges of the cake so they are even. (You can save these scraps to snack on or serve over ice cream.) Using an offset spatula or small rubber spatula, spread the melted chocolate evenly over the top and onto the sides of the cake.
Refrigerate the cake until the chocolate is set, about 30 minutes. Fill a tall glass with hot water and dip a chef's knife inside. Dry off the knife then cut the cake into 2-inch squares, warming and drying the knife before each cut to minimize the chocolate cracking.
