2 cups flaked sweetened coconut

1 (3.5 ounce) package instant coconut cream pudding mix
1 (14 ounce) can coconut milk
1 (18.25 ounce) package white cake mix
1 cup vanilla yogurt
1/4 cup vegetable oil
3 eggs
1/2 cup flaked sweetened coconut

Frosting:
1 (8 ounce) package cream cheese, at room temperature
1 cup confectioners' sugar, or to taste
2 tablespoons milk
1 teaspoon vanilla extract