Begin by preheating your oven to 350°F (175°C). Prepare a muffin tin by filling it with paper liners or lightly greasing each cup to prevent sticking.
In a large mixing bowl, mix together the rolled oats, freshly grated carrots, all-purpose flour, packed brown sugar, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt. Stir until all dry ingredients are well combined.
In a separate medium bowl, whisk together the unsweetened applesauce, milk, large egg, and vanilla extract until the mixture is smooth and homogenous.
Pour the wet mixture into the bowl of dry ingredients. Gently mix with a spatula or wooden spoon until everything is just combined. Avoid overmixing; it's perfectly fine if there are some lumps. If you're using them, incorporate the chopped walnuts or raisins at this stage.
Carefully divide the batter among the prepared muffin cups, filling each about two-thirds full to allow room for rising.
Place the muffin tin in the preheated oven and bake for 20 to 25 minutes. The muffins are done when a toothpick inserted into the center of one comes out clean.
Once baked, allow the muffins to cool in the pan for about 5 minutes. Then, transfer them to a wire rack to cool completely.
Notes
Enjoy these muffins warm or with a dollop of yogurt or nut butter.