Preheat your oven to 350°F (175°C). Grease and line a 9x13-inch baking pan with parchment paper, leaving some overhang for easy removal.
In a large bowl, combine the grated carrots and zucchini.
In another bowl, whisk together the flour, oats, baking powder, baking soda, cinnamon, ginger, and salt.
In a separate mixing bowl, beat together the granulated sugar, brown sugar, vegetable oil, eggs, and vanilla extract until well combined.
Gradually add the dry ingredients to the wet mixture, stirring until just combined. Fold in the grated carrots and zucchini.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 25-30 minutes or until a toothpick inserted in the center comes out clean. Allow the bars to cool completely in the pan.
While the bars are cooling, prepare the frosting by beating together the softened cream cheese and butter until smooth.
Gradually add the powdered sugar, lemon juice, and lemon zest, mixing until well combined and creamy.
Once the bars are completely cooled, spread the lemon cream cheese frosting evenly over the top. Cut into squares and serve.