Preheat oven to 350°F. Spray a 9x9-inch pan with nonstick cooking spray.
Whisk baking powder, salt, cinnamon, ginger, and flour in a medium bowl. Set aside.
Cream brown sugar and butter in the bowl of an electric mixer fitted with a paddle attachment. Add egg and vanilla and beat until combined. Mix in dry ingredients. Stir in carrots.
Press into prepared pan (dough is sticky, spray your hands with cooking spray to avoid it sticking to your hands).
Bake for about 25-30 minutes. (Be careful not to overcook - the center will still be jiggly when you take these out of the oven and they will finish cooking and firming up as they cool. They stay super soft though!)
Cool completely in pan before frosting.
Make your frosting: beat cream cheese and butter together with an electric mixer. Slowly mix in powdered sugar. Add vanilla and cinnamon and mix until smooth. Frost bars in pan.
Notes
Note: be sure to shred your own carrots and don’t pack them in the measuring cup.