Prep: Preheat your oven to 350°F (175°C). Line an 8x8-inch baking dish with parchment paper (leaving an overhang for easy removal) or spray thoroughly with non-stick cooking spray.
Mix Filling: In a large mixing bowl, use an electric hand mixer to beat the room-temperature cream cheese, egg, and vanilla extract until completely smooth and lump-free.
Base Layer: Cut the roll of sugar cookie dough directly in half. Press one half into the bottom of the prepared baking dish, creating an even layer that reaches the edges.
Add Cheesecake: Pour the smooth cream cheese mixture over the sugar cookie crust, using a spatula to spread it evenly.
Top Layer: Break the remaining half of the sugar cookie dough into small pieces. Flatten them slightly with your fingers to make small "chunks." Place these flattened chunks evenly across the top of the cream cheese filling.
Bake: Bake on the center rack for 30 minutes, or until the top cookie layer is lightly golden and the cheesecake center is set.
Cool & Top: Remove from the oven and allow to cool completely at room temperature for at least 1-2 hours. Once fully cooled, scatter rainbow sprinkles generously over the top.
Slice: For the cleanest slices, chill the cooled bars in the refrigerator for an additional 2 hours before cutting with a hot, clean knife. Serve and enjoy!