Lightly grease a standard 12-cup muffin tin with oil then line the cups with paper muffin wrappers. Set aside.
In a large bowl, whisk together the flour, cinnamon, baking powder, baking soda, and salt.
Add the grated carrots and apple, coconut, raisins, and seeds to the flour mixture. Use a rubber spatula or wooden spoon to stir everything together, breaking up any larger clumps of grated carrot or apple as needed.
In a second bowl, whisk together the sourdough starter, oil, sugar, eggs and vanilla until smooth.
Make a well in the carrot-flour mixture and pour in the wet ingredients. Again, using a rubber spatula or wooden spoon, fold everything together until just combined with a few streaks of flour remaining.
Divide the muffin batter evenly between the cups in the prepared muffin tin. Top the muffins with a little coarse sugar if you'd like.
Place the tin in the preheated oven and bake the muffins for 10 minutes.
Reduce the heat to 375ºF (190ºC) and continue to bake the muffins for 10 minutes more.
Remove the muffins from the oven, placing the tin on a wire rack to cool for 10 to 15 minutes.
Transfer the muffins from the tin to the wire rack and let them cool completely before storing them an airtight container or zip-top plastic bag.