Preheat the oven to 375 degrees F. Line a muffin pan with 18 paper liners. Spray liners with non-stick spray or coat with a thin layer of butter.
Lay a cheesecloth or tea towel in a small bowl and toss the shredded carrots in. Sprinkle the 1/2 teaspoon of salt over the shredded carrots and massage it into the carrots. Cover and leave it to rest for about 20 minutes. After resting, twist and squeeze the towel with the carrots in it, getting as much juice as you can out of the carrots. Then, brush the carrots off the towel and into the bowl and set aside.
In a medium bowl, combine the flour, baking powder, baking soda, salt, cinnamon, allspice, and nutmeg, and whisk together.
In another large bowl, combine sourdough starter, eggs, oil, vanilla extract, brown sugar, granulated sugar, and buttermilk* and mix well.
Add the dry ingredients to the wet ingredients, gently mixing until most of the flour is combined. Use a spatula to fold in the grated and squeezed carrots. Optional at this point, you may fold in 1/3 cup of chopped walnuts.
Spoon enough batter into each liner, filling about 2/3rds full. Bake for 18-20 minutes or until the tops are springy and a toothpick inserted into the center comes out clean or with just a few crumbs. Cool muffins in pan for 5 minutes and then remove to a wire rack to cool completely.