Preheat the oven to 350°F (175°C). Line a standard 12-cup muffin pan with paper liners or lightly grease the wells.
Wash and grate the carrots, zucchini, and apple. Gently squeeze excess moisture from the grated zucchini using a clean kitchen towel.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, and salt.
In a separate large bowl, whisk the egg and brown sugar. Add applesauce, Greek yogurt, and vanilla extract, mixing until smooth.
Add the dry ingredients to the wet ingredients and gently fold until just combined.
Fold in the grated carrots, zucchini, and apple until evenly distributed.
Divide the batter evenly between the 12 muffin cups, filling each approximately two-thirds to three-quarters full.
Bake for 20–24 minutes, or until the tops are set and a toothpick inserted into the center comes out clean or with a few moist crumbs.
Allow the muffins to cool in the pan for five minutes, then transfer to a wire rack and cool completely.
For the frosting, beat the softened cream cheese until smooth. Add powdered sugar and vanilla extract, mixing until creamy.
Spread an even, thin layer of frosting over each completely cooled muffin.
Refrigerate frosted muffins in an airtight container and serve as desired.