Preheat oven to 350°F (180°C). Grease a 13- x 9-inch baking pan and line with parchment paper.

Heat 2 tablespoons butter in a skillet over medium heat and sauté carrots until softened and slightly browned, about 8 minutes.

Transfer cooked carrots to a bowl, mix with cardamom, and refrigerate uncovered until cooled, about 15 minutes.

In a large bowl, combine spice cake mix, 1 egg, 1 1/2 teaspoons vanilla extract, and remaining butter. Mix until even, about 1 minute.

Spread the mixture evenly in the prepared pan.

In a separate bowl, beat cream cheese until smooth, then mix in powdered sugar, lemon zest, salt, remaining eggs, and vanilla until creamy, 2 minutes.

Fold in chilled carrots and pecans, then spread this cream cheese mixture over the cake layer.

Bake in preheated oven until lightly golden and crackly, about 40 minutes.

Cool completely in the pan on a wire rack, about 2 to 3 hours.