Special equipment: a piping bag with a medium star tip
Preheat the oven to 350 degrees F.
Cut 3 pieces of parchment paper to fit the bottom of a 9-by-13-inch metal baking pan. Spray the pan with cooking spray, line it with 1 piece of parchment, and spray the parchment. Reserve the other 2 parchment pieces.
Prepare the cake mix batter according to the package directions. Transfer the batter to a large liquid measuring cup and then divide it equally among 3 bowls. Use red food coloring to tint each bowl a different shade of pink: about 4 drops for lightest color, 8 for medium, and 16 for the darkest.
Pour the darkest batter into the prepared baking pan and smooth it with an offset spatula. Shake the pan vigorously and tap it against the counter a few times to even out the batter and release any air bubbles. Bake until set and the edges are starting to pull away from the pan, about 9 minutes. Cool in the pan for 2 minutes, then invert onto a cooling rack and let cool completely. Invert the cake onto a large cutting board and spread it with 1/3 cup of the jam.
Meanwhile, cool the baking pan under running water, dry well, and prepare as before with cooking spray, another piece of parchment, and more spray. Repeat the baking and cooling steps with the medium-colored batter. When completely cool, flip the cake directly onto the first cake -- it's okay if the edges don't line up perfectly -- and spread with the remaining 1/3 cup jam.
Cool the pan once again and repeat with the lightest batter. When cool, flip directly on top of the layered cake.
Put the white chocolate chips in a medium bowl. Heat the cream to just shy of a boil in the microwave. Pour over the chips and let stand for 30 seconds before stirring until smooth. Refrigerate until cold, about 20 minutes. Beat the frosting with an electric mixer on medium-high speed until light and slightly whipped, about 2 minutes. Transfer to a piping bag fitted with a medium star tip.
Trim the edges of the cake and cut into 24 squares. Pipe each square with frosting to cover. Sprinkle with the freeze-dried strawberries.