1. Prepare Oven and Pan:

• Preheat oven to 350°F (175°C).
• Grease an 8×8 inch baking pan. Lining the pan with parchment paper, leaving an overhang on two sides for easy lifting, is highly recommended. Grease the parchment paper as well. Set aside.

2. Make the Carrot Cake Batter:

• Melt the ½ cup of butter in a microwave-safe bowl or small saucepan. Let it cool slightly.
• In a medium bowl, add the melted butter and the packed light brown sugar. Mix well to combine.
• Stir in the large egg and 1 tablespoon of vanilla extract until incorporated.
• Add the all-purpose flour, ground cinnamon, baking powder, and salt to the bowl. Stir until completely combined, but do not overmix.
• Gently fold in the shredded carrots. Set the carrot cake batter aside.

3. Make the Cheesecake Swirl Mixture:

• In a separate medium bowl, add the room temperature cream cheese and granulated sugar.
• Beat with a hand mixer (or vigorously with a whisk) until completely smooth and no lumps remain.
• Add the egg yolk and ¾ teaspoon vanilla extract. Beat again until smooth and creamy.

4. Assemble and Swirl the Bars:

• Add about half of the carrot cake batter into the prepared 8×8 pan. Spread it out into an even layer using an offset spatula or the back of a spoon.
• Dollop about half of the cheesecake mixture randomly over the carrot cake batter layer.
• Gently spread the remaining carrot cake batter over the cheesecake dollops and around them, trying to cover most of the cheesecake layer (it’s okay if it’s not perfect).
• Add the remaining cheesecake mixture by dropping dollops randomly on top of the final carrot cake batter layer.
• Using a knife or a skewer, gently swirl the two batters together. Create figure-eights or simple lines through the batters. Swirl as much or as little as you like, but avoid over-mixing them into one uniform color.

5. Bake:

• Place the pan in the preheated oven.
• Bake for about 35-40 minutes. The edges should be golden brown, and the center should have very little jiggle when gently nudged. A toothpick inserted into a cakey part (not a cheesecake swirl) should come out clean or with moist crumbs.

6. Cool Completely:

• Let the bars cool completely in the pan on a wire rack before cutting. This is important for the cheesecake swirl to set fully and for clean slices. Chilling them in the refrigerator for an hour after cooling can make cutting even neater.

7. Slice and Store:

• Once completely cool (and potentially chilled), use the parchment paper overhangs to lift the bars out of the pan onto a cutting board.
• Cut into squares or rectangles.
• Store leftovers in an airtight container in the refrigerator.