Place milk in a medium saucepan set over medium heat. Bring just to a boil, and remove from heat. Stir in shortening, sugar, and potatoes. Let sit until lukewarm.
In the bowl of an electric mixer fitted with the paddle attachment, place yeast and 1/4 cup warm water (105 degrees to 115 degrees). Beat on low speed to combine. Turn off mixer, and let stand until creamy, about 5 minutes.
Place flour, salt, and baking powder in a medium bowl; whisk to combine, and set aside. Spray a large bowl with cooking spray; set aside.
Add the milk mixture to the yeast mixture, and beat on the lowest speed until combined. Beat in egg and vanilla. Add dry ingredients, and beat until moistened. Switch to next highest speed, and beat until a soft, smooth dough forms, about 5 minutes. Transfer to prepared bowl. Cover with plastic wrap, and let dough rise until it doubles in bulk, about 2 hours.
Spray two sheets of parchment with cooking spray. Roll the dough out between the sheets to a 1-inch thickness. Transfer to refrigerator, and let rest for 20 minutes.
Remove top layer of parchment. Using a 2 1/2-inch round cutter, cut out doughnuts as close together as possible, dipping cutter in flour before each cut. Use a 3/4-inch round cutter to cut a hole in center of each doughnut.
Line a baking sheet with paper towels; set aside. Heat 4 inches of oil in a large stockpot to 375 degrees. Working in batches so as not to crowd the pan, fry doughnuts until golden brown, about 1 1/2 minutes per side. Use a slotted spoon to transfer to prepared baking sheet to drain and cool. Dip the top of each doughnut in glaze, alternating between the maple-orange and chocolate flavors. Coat glaze with sprinkles, if desired. Turn right side up to air dry.