Heat oven to 350°F. Grease 2 large baking sheets with solid vegetable shortening. In a medium-size bowl, whisk together flour, baking soda and salt. Set aside.
In a large bowl with an electric mixer, beat together sugar, butter and eggs until light and fluffy, about 2 minutes. In a small bowl or measuring cup, whisk together sour cream, orange juice and orange zest. On low speed, add flour mixture to sugar mixture, alternating with the sour cream mixture, ending with the flour.
Drop 1/4-cupfuls of batter onto prepared sheets, spacing about 3 inches apart. Spread batter slightly into 3-inch rounds. Transfer cookie-filled sheets to 350°F oven. Bake for 13 to 15 minutes or until cookies are puffed and lightly browned, and spring back slightly when pressed. Remove to a wire rack to cool completely. Repeat with remaining batter.
While cookies cool, prepare Icing. In medium bowl, combine confectioners sugar, milk and vanilla extract. Transfer 3/4 cup icing to a small bowl. Stir in cocoa powder and water. Transfer 1/3 cup of the remaining white icing to a second small bowl; tint with red food coloring. If needed, thin white icing with 1/2 teaspoon water.
Ice half of each cookie with the chocolate icing. Let dry 5 minutes. Frost second half of cookie with either red or white icing. Let all cookies dry for 45 minutes at room temperature. Store iced cookies in a single layer in 1 or 2 airtight containers at room temperature for up to 5 days.