Combine mangoes, peaches, sugar, and cornstarch in a medium saucepan. Cook the mixture over medium heat and bring to a boil. Stir often until the fruits are soft and covered with a thick gel-like liquid. Set aside to cool.
Let the dough sit at room temperature for 5 minutes. On a floured board, roll it out to 1/4 inch thickness*. Using cookie cutter that is 3.5 inches in diameter, cut as many circles as you can from the dough.
To fill the dough, roll the rolling pin gently over one circle to elongate it a bit more. Spoon about a tablespoon of the filling into the center of the dough. Fold the dough over, covering the filling completely. Seal the ends and use the tines of a fork to press some decorations along the edges*. Repeat with the rest of the circles. Arrange all assembled pies on a baking sheet that is lined with parchment paper. Chill them in the fridge as you heat up the oil for frying.
Fill a heavy-bottomed saucepan with the shortening and melt over medium heat. As it melts, it should occupy about 3-4 inches of the height of the saucepan. Add more shortening as needed. Once the temperature reaches 365 F, gently place 3-4 pies in the saucepan.
Cook until the pies are lightly golden, about 2-3 minutes or less. Drain the cooked pies on a plate lined with paper towels. Repeat with the remaining pies, carefully monitoring the temperature of the oil to maintain the proper heat. If the pies brown too quickly, your oil may be too hot. Adjust the heat accordingly, as you fry all the remaining pies.
Dust pies with powdered sugar and serve warm. Store leftover pies in a paper box container or bakery paper bags at room temperature for up to 2 days.