Preheat the oven to 350°F. Line a 9x13-inch pan with parchment paper so two opposite sides overlap to form handles; lightly spray the parchment with non-stick spray and set the pan aside.
In a large bowl combine 1/2 cup packed light brown sugar, 1/2 cup granulated sugar, 1 cup very ripe mashed banana, 1 teaspoon pure vanilla extract, 1/3 cup canola oil (or melted coconut oil), and 2 large room-temperature eggs. Whisk until smooth and well combined.
Add the dry ingredients—1 cup all-purpose flour, 1 teaspoon baking powder, 1/2 teaspoon baking soda, 3/4 teaspoon cinnamon, 1/4 teaspoon ground nutmeg, and 1/4 teaspoon salt—to the wet mixture. Stir with a spatula or spoon until just combined; do not overmix.
Pour the batter into the prepared pan and spread it into an even layer with a rubber spatula.
Bake for 20–22 minutes, or until a toothpick inserted in the center comes out clean or with just a few moist crumbs attached.
Remove the pan from the oven and place it on a cooling rack. Let the bars cool completely in the pan before frosting.
While the bars are cooling, make the frosting: in the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using an electric handheld mixer, beat 8 ounces very soft full-fat cream cheese, 2 tablespoons very soft unsalted butter, and 2 teaspoons pure vanilla extract on medium-high speed until completely smooth (about 2 minutes).
Reduce the mixer speed to low and gradually add 2 cups sifted confectioners' sugar. After all the sugar is incorporated, beat on high speed for 1–2 minutes until the frosting is smooth and spreadable.
Once the bars are completely cool, use the parchment overhang to lift them out of the pan and set them on a cutting board or serving tray. Gently spread the cream cheese frosting evenly over the top.
Cut into squares and serve. Store the frosted bars covered in the refrigerator for up to 2 days.