Coconut Layer Cake

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For the cake: 

Nonstick cooking spray

3 cups all-purpose flour, plus more for dusting

1 teaspoon ground ginger

1 teaspoon baking powder

1/2 teaspoon baking soda

1/2 teaspoon kosher salt

1/2 cup cream of coconut 

1/2 cup whole milk 

3 sticks unsalted butter, at room temperature

1 1/4 cups granulated sugar 

5 large eggs, at room temperature

1 teaspoon pure vanilla extract

1 teaspoon coconut extract

For the frosting:

2 cups sweetened shredded coconut

8 ounces cream cheese, at room temperature 

3 sticks unsalted butter, at room temperature

6 cups confectioners' sugar 

1 teaspoon coconut extract