Preheat oven to 350 degrees F (175 degrees C). Grease a 13x18-inch rimmed baking sheet.
Combine white sugar, 1 cup butter, 8 ounces cream cheese, and egg in a large bowl; beat with an electric mixer until well mixed, about 4 minutes. Beat in 1 teaspoon vanilla extract and almond extract. Mix in flour, baking powder, and baking soda until dough comes together.
Press dough evenly over the greased baking sheet.
Bake in the preheated oven until lightly golden, about 20 minutes. Let cool completely, 10 to 20 minutes.
Beat 1/2 cup butter and 4 ounces cream cheese together in a bowl until smooth. Beat in confectioners' sugar. Thin icing with milk and vanilla extract. Split icing between 2 bowls. Mix green food coloring into 1 bowl.
Spread green icing over cooled cookies. Cut into 48 squares. Pour white icing into a resealable plastic bag. Cut off corner tip. Drizzle white icing over cookies.