Prep the pan and oven. Heat the oven to 350°F (175°C).
Line a 12-cup muffin tin with paper liners or grease well. For tiny hands, you can also use a mini muffin pan and adjust the bake time.
Grate the carrots. Use the fine holes of a box grater so the pieces are soft and blend into the batter. Lightly press out excess moisture with a paper towel if the carrots are very wet.
Whisk the wet ingredients. In a large bowl, whisk the applesauce, oil (or melted butter), maple syrup or honey, egg, and vanilla until smooth and glossy.
Mix the dry ingredients separately. In a medium bowl, stir together the flour, baking powder, baking soda, cinnamon, and salt to evenly distribute the leavening.
Combine gently. Add the dry mixture to the wet mixture and stir with a spatula until just combined.
Fold in the grated carrots and any add-ins. Stop as soon as there are no streaks of flour.
Fill the cups. Divide the batter evenly among the muffin cups, about 3/4 full. For mini muffins, fill nearly to the top.
Bake until set. Bake standard muffins for 16–20 minutes, mini muffins for 10–12 minutes.
They’re done when the tops spring back lightly and a toothpick comes out clean or with a few moist crumbs.
Cool and serve. Let muffins cool in the pan for 5 minutes, then transfer to a rack. Serve slightly warm or at room temperature for the softest texture.