Carrot Cake with Ginger Cream Cheese Frosting

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For the cake: 

Cooking spray

2 3/4 cups chopped walnuts

2 1/2 cups all-purpose flour

2 teaspoons baking powder

1 teaspoon baking soda

2 teaspoons ground ginger

1 teaspoon ground cinnamon

1/4 teaspoon ground allspice

1/4 teaspoon ground nutmeg

1/4 teaspoon salt

1 1/2 cups granulated sugar

1/2 cup packed light brown sugar

1 1/4 cups vegetable oil

4 large eggs

1/2 cup smooth applesauce

Finely grated zest of 1 orange

1 teaspoon pure vanilla extract

2 cups grated carrots (about 4 medium)

For the frosting:

1 4-inch piece ginger

1 tablespoon fresh orange juice

1/2 teaspoon pure vanilla extract

2 8-ounce packages cream cheese, at room temperature

1 stick unsalted butter, at room temperature

3 cups confectioners' sugar, sifted