Heat oven to 350°F (175°C). Grease and line an 8×8-inch baking pan with parchment paper (leave an overhang so you can lift the bars out easily).
Finely grate 1 ½ cups of fresh carrots. (Freshly grated gives the best texture—skip the pre-shredded bagged kind, they’re too dry.)
In a large mixing bowl, whisk together 3 eggs, ½ cup vegetable oil, ½ cup applesauce (or crushed pineapple), ¾ cup sugar, and 1 teaspoon vanilla extract until well combined and slightly frothy. This is the base that makes the bars soft and fluffy.
Sift or sprinkle in 2 cups all-purpose flour, 2 teaspoons baking powder, ¼ teaspoon nutmeg, ½ teaspoon ground ginger, 1 ½ teaspoons cinnamon, and ¼ teaspoon salt. Gently stir with a spatula until just combined. (Don’t overmix—this keeps the bars tender.)
Add the shredded carrots and fold through the batter until evenly distributed. The batter will be thick and spiced—it already smells amazing!
Pour the batter into your prepared pan and smooth the top. Bake for 35–40 minutes, or until the top is golden and a toothpick inserted in the center comes out clean. Let cool completely in the pan.
In a medium bowl, beat together 8 oz softened cream cheese, 3 tablespoons unsalted butter, ½ cup powdered sugar, 1 teaspoon vanilla extract, and ¼ teaspoon ground cinnamon until smooth, creamy, and fluffy (2–3 minutes).
Spread the frosting generously over the cooled carrot cake bars. Dust lightly with extra cinnamon if you like, then slice into squares.