Preheat oven to 350°F (180°C). Spray an 8-inch square baking pan with cooking spray. Line pan with parchment paper, letting excess extend over sides of pan.
In the bowl of a stand mixer fitted with the paddle attachment, beat brown sugar, oil, eggs, and vanilla at medium-low speed until well combined, about 1 minute, stopping to scrape sides of bowl.
In a medium bowl, whisk together flours, cinnamon, salt, baking powder, ginger, nutmeg, and baking soda. With mixer on low speed, gradually add flour mixture to sugar mixture, beating until barely combined. Add carrot and pineapple; beat just until combined, stopping to scrape sides of bowl. Pour batter into prepared pan, smoothing into an even layer with a small offset spatula. Tap pan on a kitchen towel-lined counter a few times to settle batter and release any air bubbles.
Bake until a wooden pick inserted in center comes out clean, 20 to 26 minutes. Let cool in pan for 10 minutes. Using excess parchment as handles, remove from pan, and let cool completely on a wire rack.
Spread Cream Cheese-Yogurt Glaze onto cooled cake. Garnish with cinnamon, if desired. Serve immediately.