Preheat oven to 350°F. Line an 8×11- inch pan with parchment paper, leaving an overhang on the sides to make for easy removal.
In a medium-sized bowl, mix together flour, baking powder, salt, cinnamon, ginger, nutmeg, and cloves. Set aside.
In a small pot over medium heat, melt the butter. Once melted, reduce heat to medium-low and continue cooking and stirring frequently until butter turns golden and smells nutty, about 5-10 minutes. Watch closely, you don't want it to burn! Remove from heat and transfer to a microwave-safe bowl. Let cool for a few minutes.
Once the browned butter has slightly cooled, whisk in brown sugar and vanilla. Add in the whole egg and egg yolk and combine until fully incorporated. Fold the carrots in and then slowly add the dry ingredients to the wet ingredients until just combined. The dough will be very thick and pasty. Spread the dough evenly onto your parchment-lined pan. Bake for 25-30 minutes or until browned on top. Let blondies cool completely.
While the blondies are baking, make the cream cheese frosting by combining the cream cheese and butter in a stand mixer with the paddle attachment on medium speed until smooth. Add the powdered sugar, vanilla, and salt and beat on low speed for a minute, then increase to high speed for 3 minutes until completely combined.
Once blondies have cooled, spread frosting on top in an even layer. Store in the refrigerator or serve.
Notes
When you're browning the butter, don't go far from the pot. Watch closely for the little brown bits to start forming (they're called milk solids). Once those become brown, remove the pot from the heat and transfer to a microwave-safe bowl.
An extra egg yolk goes a long way to make the blondies EXTRA fudgy. It will make the blondies have that slightly undercooked texture, extra fudgy and chewy.
The dough should be thick and pasty. Spread it as evenly as possible across the baking pan.