Make the carrot bread: Preheat the oven to 350 degrees F. Lightly butter one 9-by-5-inch loaf pan or three 6-by-3-inch mini loaf pans.
Whisk the flour, granulated sugar, baking powder, salt, baking soda, cinnamon and nutmeg in a large bowl. Add the hazelnuts and coconut.
Whisk the eggs, melted butter, yogurt, vanilla and lemon zest (if using) in a medium bowl. Stir in the shredded carrots. Fold the carrot mixture into the flour mixture until just combined.
Spread the batter in the prepared pan(s). Bake until a toothpick inserted into the center comes out clean, about 55 minutes for a standard loaf or 35 to 40 minutes for mini loaves. Cool 30 minutes in the pan(s) on a rack, then turn out onto the rack to cool completely.
Make the glaze: Whisk the confectioners' sugar, cream cheese, milk and vanilla in a bowl. Pour over the cooled carrot bread and let set, 15 to 20 minutes.
Photograph by Charles Masters
Recipe courtesy Food Network Magazine