Preheat oven to 375°F. Line a 9-inch-square baking pan with parchment paper, allowing the parchment to extend over the sides by at least 2 inches.

Whisk 1¼ cups oat flour, ¾ teaspoon baking powder, ¾ teaspoon salt, ½ teaspoon cinnamon and ¼ teaspoon cloves together in a small bowl.

Place 5 tablespoons butter in a large microwave-safe bowl; microwave on High just until melted, about 30 seconds.

Whisk in ¾ cup brown sugar until combined. Add 1 egg, 1 egg yolk and ½ teaspoon vanilla; whisk until combined.

Add the flour mixture and whisk just until the dry ingredients are moistened. Fold in 1 cup carrot.

Transfer the batter to the prepared baking dish. Bake until golden brown and slightly puffed and a toothpick inserted in the center comes out almost clean with a few moist crumbs, 25 to 27 minutes.

Let cool completely in the pan on a wire rack, about 1 hour.

Meanwhile, beat 3 ounces cream cheese and ⅓ cup yogurt in a medium bowl with an electric mixer on medium speed until smooth and creamy, about 30 seconds, scraping down the sides and bottom of the bowl as needed.

Add 2 tablespoons confectioners’ sugar and the remaining ½ teaspoon vanilla; beat until completely smooth, about 30 seconds.

Use the parchment paper to remove the cooled bars from the pan. Place the bars on a plate or platter; discard the parchment.

Spread the frosting evenly over the bars and top with 2 tablespoons walnuts. Cut into 12 pieces.