Preheat the oven to 170°C / 340°F.
In a mixing bowl, whisk together the brown sugar, eggs, vegetable oil, and vanilla extract until smooth.
Gently fold the shredded carrots into the wet mixture.
In a separate bowl, combine the flour, baking soda, cinnamon, and salt.
Gradually add the dry ingredients to the wet mixture, stirring until no dry flour remains.
Divide the batter between two greased and lined 8-inch pans.
Bake for 30–40 minutes, or until a cake tester comes out clean.
Let the cakes cool in the pans for 10 minutes, then transfer to a wire rack.
Add the cream cheese and mix until smooth.
Gradually incorporate the powdered sugar until fully combined.
Frost the cooled cakes, adding a layer of frosting between them.
Chill for 30–60 minutes before serving.