Step 1Heat the oven to 375 degrees. In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using a hand mixer, cream the butter and shortening. Add the salt and sugar, and continue to beat until the mixture is light and fluffy, 3 to 5 minutes. Add the egg and beat well. Beat in the orange extract, then the flour. Mix in the coconut and chopped nuts by hand.
Step 2Drop rounded tablespoonfuls of dough onto an ungreased cookie sheet, spacing the cookies 2 inches apart. Bake until the edges are lightly browned, 10 to 14 minutes, rotating the cookies halfway through for even baking.
Step 3Cool the cookies on the pan for 1 minute before removing. Sprinkle with sugar and set aside to cool completely before frosting.
Step 1In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl using a hand mixer, slowly beat the butter into the powdered sugar. Beat in the orange juice, then the lemon juice, along with the lemon zest. Beat in additional powdered sugar, a tablespoon at a time, to thicken the frosting to achieve the desired consistency. This makes about three-fourths cup frosting.