Prep: Preheat oven to 350°F. Grease a 9x13 metal pan, line with parchment (leave an overhang), and grease the paper.
Puree: Steam carrots until very tender (20–25 mins). Puree while hot with 2 tbsp butter until smooth. Set aside to cool.
Brown Butter: Melt 12 tbsp butter in a saucepan over medium heat. Cook until golden brown and nutty (5–6 mins). Remove from heat, stir in remaining 4 tbsp butter until foaming stops. Whisk in brown sugar, vanilla, and salt. Cool to room temperature.
Cheesecake Mix: Whisk softened cream cheese, sour cream, granulated sugar, vanilla, and egg until smooth.
Combine Batter: Whisk cooled carrot puree into the brown butter mixture. Whisk in eggs and yolks. Fold in dry ingredients, then the pecans.
Layer & Swirl: Spread half the carrot batter in the pan. Dollop half the cream cheese mix on top and swirl with a knife. Cover with remaining carrot batter, top with the rest of the cream cheese, and swirl again vigorously.
Bake: Bake for 30–35 minutes until a toothpick comes out with just a few crumbs. Cool completely in the pan on a wire rack before lifting out and slicing.