How to Make It
Beat mashed potatoes and sugar with an electric stand mixer on low speed until well combined. Add eggs, and beat until combined. Add milk and shortening, and beat until well combined.
Replace beater with dough hook attachment. Whisk together flour, baking powder, and salt in a large bowl; gradually add in batches to potato mixture, and beat on low until blended after each addition, stopping to scrape down sides when necessary.
Turn dough out on a lightly floured surface, and knead 3 to 4 times. Divide dough in half, and roll each portion to 1/4-inch thickness. Using a floured 2 1/2-inch round cutter, cut out 40 circles. Place about 1 teaspoon of cranberry sauce in the center of 20 circles; brush edges with egg white. Top with remaining 20 circles; press edges to seal tightly.
Pour oil to the depth of 3 inches in a large Dutch oven over medium (about 360°F). Gently place doughnuts in hot oil, and cook in batches until golden brown, about 2 minutes on each side. Transfer to a plate lined with paper towels. Sprinkle with powdered sugar, and serve immediately.