Preheat oven to 350°F. Line a 9×5-inch loaf pan with parchment paper, allowing it to hang over the sides for easy removal. Set aside.
In a large bowl, whisk together oil, brown sugar, and granulated sugar.
Add eggs, one at a time, mixing well after each addition.
Stir in the sour cream and pineapple. Whisk to combine.
Add vanilla extract. Stir until well combined.
In a separate bowl, whisk together the flour, baking powder, cinnamon, baking soda, salt, and nutmeg.
Gradually add the dry ingredients to the wet ingredients. Mix until just combined (do not overmix).
Gently fold in the grated carrots.
Pour the batter into the prepared loaf pan, smoothing the top with a spatula.
Bake for 55 to 60 minutes, or until the top is golden brown and a toothpick inserted into the center of the loaf comes out with a few crumbs, but no wet batter. If the top starts to brown too quickly, loosely tent the loaf with aluminum foil.
Once done baking, remove the loaf from the oven and let cool in the pan for about 30 minutes. Then, transfer to a wire rack to cool completely.