Using a box grater or food processor, grate the zucchini and carrots. Stir 2 tablespoons sugar into shredded veggies, and place mixture in a large colander set over a bowl. As you prepare the rest of the batter ingredients, the sugar will draw the moisture out of the zucchini and carrots. Occasionally press down on them to squeeze more water into the bowl.
Preheat the oven to 375°F. Lightly coat two mini muffin pans with cooking spray.
Place chopped nuts on a sheet pan and bake for 5 to 7 minutes or until toasted. Allow to cool for a few minutes.
In a large bowl, whisk together nuts, flour, baking soda, baking powder, and salt. In a medium bowl, stir together coconut oil or butter, yogurt, eggs, lemon juice, and vanilla. Blend in sugar.
Add liquid mixture to flour mixture and stir until just combined. Press on shredded zucchini and carrots in colander one last time to extract any remaining water. Mix veggies into batter.
Scoop batter into mini muffin tins and bake for 10 minutes, or until a toothpick inserted in the center of a muffin comes out clean. You will probably have more batter than will fit in two muffin pans. You may bake the remaining batter in an additional mini muffin pan (or reuse a pan after the first batch of muffins comes out of the oven, allowing it a few minutes to cool off). Or you can scoop the remaining batter into small ramekins to make a couple of larger muffins (which will probably take 15 minutes or so to bake).
Notes
I had a 10 ounce zucchini on hand, so I used 6 ounces of carrots in addition to that in order to make up my pound of veggies.
You can use all-purpose white flour if you prefer. You can also use half white flour and half whole wheat flour in lieu of whole wheat pastry flour.
If you wish to use unrefined sugar, Sucanat and Rapadura are popular brands.
If you wish to make regular-sized muffins instead of mini, this recipe will probably yield 16 to 18 regular muffins. Start checking them for doneness at 15 minutes.