Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking pan or line it with parchment paper.
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, eggs, vegetable oil, 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 until evenly distributed.
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 on a wire rack.
While the bars are cooling, prepare the frosting. In a mixing bowl, beat the softened cream cheese and butter together until smooth.
Gradually add the powdered sugar, lemon juice, and lemon zest, mixing until creamy and well combined.
Once the bars are completely cool, spread the lemon cream cheese frosting evenly over the top. Cut into squares and serve.