Preheat the oven to 350°F. Grease a 9x13-inch baking pan or line it with parchment.
In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon and salt.
In a large bowl, whisk the eggs, oil, brown sugar, granulated sugar and 1½ teaspoons of the vanilla until smooth.
Stir the dry ingredients into the wet until just combined, then fold in the grated carrots.
Spread the batter in the pan. Bake for 25–30 minutes, until a toothpick in the center comes out clean.
Cool completely in the pan.
For the frosting, beat the cream cheese and butter until smooth. Beat in the powdered sugar and the remaining ½ teaspoon vanilla until creamy.
Spread the frosting over the cooled bars and cut into 16 squares.