Prepare carrots and the pan. Heat a conventional oven to 375°F (190°C). Line twelve standard muffin cups and place a rack in the center of the oven. Finely grate and weigh the carrots, then chop the walnuts into small even pieces. Prepare these ingredients now so the mixed batter does not have to wait while you finish grating.

Mix the dry ingredients. Whisk flour, both sugars, baking powder, baking soda, cinnamon, nutmeg, and salt in a large bowl. Break up brown sugar lumps with your fingertips or the back of a spoon. Whisk along the bottom of the bowl to distribute the cinnamon and raising agents rather than leaving them in a surface layer.

Combine eggs and carrot. In another bowl, whisk the eggs, milk, oil, and vanilla until evenly combined. Stir in the grated carrot so the strands are separated and coated in the wet mixture. Do not drain off the carrot juices or add extra milk simply because the mixture looks textured; the grated vegetable is meant to remain visible.

Fold without beating. Pour the carrot mixture into the dry ingredients and fold gently until a few flour streaks remain. Add the chopped walnuts, then finish folding just until no dry pockets remain. The batter should be thick and scoopable. Divide evenly among the twelve prepared cups, keeping walnut pieces distributed rather than leaving them all at the bottom of the bowl.

Bake until set. Bake for about 22–26 minutes, checking the center muffins near the start of the range. Tops should be lightly browned and spring back gently, and a tester should come out without wet batter. Carrot strands can appear on the tester without meaning the center is raw; inspect the crumb around them rather than judging color alone.

Cool and release. Leave the muffins in the pan for 5 minutes, then lift them onto a wire rack for at least another 10 minutes. Let them cool completely before sealing or freezing. If liners cling to a very warm muffin, allow more cooling time before peeling rather than pulling away a large piece of the soft base.