Preheat oven to 350°F and grease a mini muffin pan.
In a bowl, mix together the yogurt, applesauce, egg, honey, oil, and vanilla until smooth.
Add in the flour, grated carrot, cinnamon, baking powder, baking soda, and salt. Gently stir until just combined. Do not overmix.
Scoop into mini muffin pan, filling each about 3/4 full.
Bake for 14–17 minutes, or until set and lightly springy in the center.
Let cool slightly before serving.