Honey Carrot Cake

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Carrot Cake:

Unsalted butter, for the pan

1 1/4 cups all-purpose flour, plus more for the pan

1 cup whole wheat flour 

2 teaspoons baking powder 

1 teaspoon ground cinnamon 

1/2 teaspoon ground allspice 

1/2 teaspoon freshly grated nutmeg 

1/2 teaspoon fine salt 

1/2 cup vegetable oil 

1/2 cup applesauce 

1/2 cup honey 

1 1/2 teaspoons pure vanilla extract 

2 large eggs 

1 cup grated carrots (about 2 large carrots) 

1/2 cup chopped pecans 

1/2 cup shredded sweetened coconut 

Cream Cheese Glaze:

2 ounces cream cheese, at room temperature

3 tablespoons unsalted butter, at room temperature 

1/4 cup confectioners' sugar 

1/2 teaspoon pure vanilla extract 

2 to 3 tablespoons milk