Preheat the oven to 350 degrees Fahrenheit. Line a 9x9 pan with a parchment sling (this makes it much easier to remove the bars later) and spray it lightly with baking spray.
In a large microwave safe bowl, melt the butter. As soon as it is completely melted and while it is still warm, stir in the brown sugar. Whisk it together for about 2 minutes, until it’s thick and lighter in color.
¾ cups (12 tbsp) salted butter, 1 ¼ cups brown sugar
Whisk in the eggs, egg yolk, and vanilla until smooth.
1 egg + 1 yolk, 1 teaspoon vanilla
Using a rubber spatula, stir in the grated carrot.
¾ cups (~ 3 oz) carrot
Add the flour, salt, cinnamon, nutmeg, and cloves. Stir until the dough is almost combined, with only a few white streaks remaining.
1 ¼ cups all-purpose flour, ¼ teaspoon kosher salt, ¾ teaspoon cinnamon, ¼ teaspoon nutmeg, ⅛ teaspoon cloves
Add the chopped nuts and continue to stir until the batter is evenly combined and no white streaks are left. Be careful not to overmix in order to avoid making them tough.
½ cup walnuts
Pour the batter into the prepared pan and bake for 20-25 minutes, until a toothpick comes out clean or with only a few moist crumbs. It will be lightly golden brown and set in the middle. Be very careful not to overbake. Cool completely in the pan.
To make the frosting, beat the butter and cream cheese on high speed with a hand mixer or stand mixer for 1-2 minutes, until it is smooth. Then add the vanilla and mix again until combined.
6 tablespoon butter, 6 oz cream cheese, ½ teaspoon vanilla
Add the powdered sugar and mix on low at first (just to avoid a mess), then gradually increase the speed to medium-high for 1-2 minutes. It should be light and fluffy.
2-3 cups powdered sugar
Spread the frosting on top of the completely cooled bars, then sprinkle very lightly with cinnamon and the crumbs of the chopped walnuts, if desired. Use the sling to lift them out of the pan, then use a long sharp knife to cut them into 16 evenly sized squares.