Prep the oven and pan: Preheat your oven to 375°F (190°C).
Line a 12-cup muffin tin with paper liners or lightly grease each cup.
Grate the vegetables: Finely grate the carrot. Grate the zucchini, then place it in a clean kitchen towel and gently squeeze out excess moisture. Don’t wring it completely dry—just remove the watery drips.
Whisk the wet ingredients: In a large bowl, whisk together eggs, oil, yogurt, brown sugar, and vanilla until smooth and slightly thick.
Combine the dry ingredients: On top of the wet mixture, add flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Lightly whisk the dry ingredients together on the surface, then fold into the wet until mostly combined.
Fold in the veggies and mix-ins: Gently fold in the carrot, zucchini, and any optional mix-ins until there are no dry streaks. The batter will be thick and slightly lumpy—perfect.
Fill the muffin cups: Divide the batter evenly among the 12 cups. They should be about 3/4 full.
Bake: Bake for 18–22 minutes, or until the tops are set and a toothpick inserted in the center comes out clean or with a few moist crumbs.
Cool: Let the muffins cool in the pan for 5 minutes, then transfer to a wire rack to cool completely.
This helps keep the bottoms from getting soggy.