Heat oven to 350°F/175°C. Line a 9-by-13-inch (23-by-33 cm) rectangular pan with parchment paper.
Grate the carrots and set aside. In a large bowl, whisk eggs, granulated sugar, brown sugar, and oil until smooth and fully emulsified.
Add flour, baking soda, cinnamon, and salt. Whisk just until no dry spots remain.
Fold in carrots, then pecans, mixing gently until evenly distributed.
Transfer batter to the prepared pan and smooth the surface. Bake for 40 to 45 minutes, until the top is deeply golden and a toothpick inserted into the center comes out with moist crumbs. Let cool completely in the pan.
Prepare the frosting by whisking butter, cream cheese, and powdered sugar until smooth and creamy.
Spread frosting evenly over the cooled cake. Refrigerate at least 2 hours, or overnight, to set before slicing.