For the cake: Preheat the oven to 350 degrees F. Line a 9-by-13-inch cake pan with aluminum foil and spray with nonstick baking spray.
Add the cake mix, 1 teaspoon of the vanilla extract and 2 of the eggs to the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed until combined, about 1 minute. Add 1 stick of the butter and mix until just combined. Spread the batter evenly in the prepared pan with an offset spatula.
Beat the cream cheese in a separate bowl of a stand mixer fitted with the paddle attachment (or clean the first bowl) on medium speed until light and smooth, scraping down the sides occasionally, about 2 minutes. With the mixer running on medium-low speed, add the remaining 2 eggs and 1 teaspoon vanilla extract. Once combined, add the remaining 1/2 stick butter and beat until smooth. With the mixer running on low speed, add the confectioners' sugar, one spoon at a time, until combined, scraping down the sides occasionally. Evenly pour the mixture over the cake layer and smooth the top with an offset spatula.
Bake until the edges are puffed and golden brown and the center is just set, 30 to 35 minutes. Let cool on a wire rack for 30 minutes, and then chill in the refrigerator at least 4 hours.
For the royal icing: Add the confectioners' sugar, meringue powder and 6 tablespoons water to the bowl of a stand mixer fitted with the whisk attachment. Whisk until stiff peaks form.
Transfer the chilled cake to a baking sheet and remove the foil. Spread an even layer of royal icing on the cake with an offset spatula and decorate with sprinkles if desired. Cut into 24 bars (4 lengthwise strips and 6 crosswise strips).
It is important that all the ingredients are at room temperature. Otherwise, the batter will be difficult to spread in the cake pan.
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