Line 9×5 loaf pan with parchment paper, leaving 1-2 in. overhang on the sides. Preheat oven to 350F. Whisk together both flours, baking powder, baking soda, salt and chai spice. Set aside.
In a stand mixer with whisk attachment (or by hand with a whisk), beat oil and sugar until smooth and well combined. Add eggs, beating well after each one until well combined. Add vanilla extract, beating until well combined.
In this order (dry ingredients, yogurt, dry, yogurt, dry), add dry ingredients to the wet alternating with the yogurt. Slowly beat each time until JUST combined. A few flour streaks is okay! Fold in finely shredded carrots. Gently pour into prepared pan.
Bake at 350F for 45 minutes. Check the topping. If its browning too quickly, top with a foil tent. Return to the oven for 10-15 minutes, checking for doneness every 5 minutes (55-60 minutes total). When a toothpick inserted into the center comes out clean or with a few moist crumbs, its done. Cool in pan for at least 30 minutes. Carefully remove from pan, using the excess parchment. Continue cooling on cooling rack for an addtional 20-30 minutes. Loaf needs to be completely cooled before frosting.