Preheat oven to 350F. Line a 9x5-inch loaf pan with parchment paper, leaving overhang on the long sides for easy removal. Lightly oil any exposed pan surface.

In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt until well combined and no lumps remain.

In a medium bowl, whisk the oil, brown sugar, and granulated sugar until the mixture looks like wet sand. Add the applesauce, vanilla, and vinegar and whisk until smooth and emulsified.

Pour the wet mixture into the dry ingredients. Stir with a rubber spatula until just combined and no dry flour remains - do not overmix. The batter will be thick.

Add the grated carrots and toasted walnuts. Fold gently until evenly distributed throughout the batter, scraping the bottom of the bowl to catch any hidden flour pockets.

Transfer batter to the prepared pan and smooth the top. Bake 48-55 minutes, until a toothpick inserted in the center comes out with just a few moist crumbs and the top springs back when lightly pressed. The loaf should be deep golden brown and pulling slightly from the pan edges.

Cool in the pan for 15 minutes, then lift out using the parchment overhang and transfer to a wire rack. Cool completely before slicing - at least 1 hour - so the crumb sets properly.