Whisk together the flour, baking powder, salt, 1 cup of the sugar and 1/2 teaspoon of the apple pie spice in a medium bowl until well mixed. Stir the remaining 1/2 cup sugar and 1 tablespoon apple pie spice together in a shallow dish. Set aside.

Whisk together the eggs, melted shortening, lemon zest and 1/2 cup seltzer water in a large bowl until smooth. Stir in the flour mixture. Add the remaining seltzer water, a teaspoon at a time, just until the dough comes together in a soft and shaggy ball.

Turn the dough out onto a generously-floured cutting board (it will be a bit sticky). Sprinkle the top with flour and pat out until it is about 3/8-inch thick. Cut out as many circles as possible using a 3-inch round cookie or biscuit cutter, then cut out doughtnut holes from the center of each with a 1-inch round cutter. Reserve the doughnuts and holes. Gather the scraps together and pat them out again to 3/8-inch thickness. Continue to cut doughnuts and doughnut holes, reworking the scraps, until all the dough is used.

Heat 2 inches of vegetable oil in a heavy-bottomed pot or Dutch oven over medium-high heat until a deep frying thermometer inserted in the oil reaches 375 degrees F. Working in batches, fry the doughnuts and holes until golden brown, 2 to 3 minutes, turning occasionally. Transfer to a paper towel-lined baking sheet to drain and cool slightly. While still warm, roll in the spiced sugar until well coated.