1 stick unsalted butter, melted and cooled (or 1/2 cup vegetable oil), plus more for the pan
1 1/4 cups all-purpose flour
3/4 cup granulated sugar
1 teaspoon baking powder
1 teaspoon salt
1/2 teaspoon baking soda
1/2 teaspoon ground cinnamon
1/2 teaspoon grated nutmeg
3/4 cup hazelnuts, toasted and chopped
1 cup sweetened shredded coconut
2 large eggs
1/2 cup plain yogurt or sour cream
1 teaspoon vanilla extract
1 teaspoon grated lemon zest (optional)
1 cup shredded carrots
1 cup confectioners' sugar
3 tablespoons cream cheese, softened
2 tablespoons milk
1/4 teaspoon vanilla extract