For the dough: In a large bowl or the bowl of a stand mixer fitted with the paddle attachment, add the yeast, either active dry or instant, to the 1/2 warm water. Add the 2 teaspoons of sugar, stir and let stand 5 minutes. Meanwhile, in a large bowl, whisk together the flour, 1/4 cup sugar, baking powder, mace and salt. Set aside.
When yeast is ready, add the shortening, egg and vanilla and mix with the paddle attachment on low speed for 1 minute (to break up the shortening). Add about 1/3 of the flour mixture and mix until combined. Add a second third of the flour mixture and again, mix until combined. Switch to the dough hook and add the remaining flour mixture. Knead with the hook until all that flour is incorporated. Now start adding flour in 1 Tbsp. increments, until the dough comes together, clears the bottom of the bowl and is becoming smooth. Knead on medium for about 2 minutes more. The dough should be smooth and but still a bit moist (but not sticky).
Place the dough in an oiled bowl, cover with plastic wrap and set-aside to rise until doubled in size (about 1 hour).
Meanwhile, make the apple filling. Place the diced apples, sugar and lemon juice into a skillet over medium heat. Cook, stirring often, until all the liquid has evaporated and the apples are softened, about 5 minutes. Remove to a bowl to cool completely before using.
When bread has finished it's first rise, remove to a lightly floured surface and roll into a 10-inch x 10-inch square. Sprinkle the apples over one half of the dough. Sprinkle the 1 Tbsp. cinnamon evenly over the apples and then sprinkle with the 1 Tbsp. flour. Fold the empty side of the dough over the side with the apples. Using a sharp knife or a bench scraper, cut the dough into 1/2-inch wide strips from top to bottom. Repeat cutting 1/2-inch strips from side to side, so you end up with small chunks of dough. Gather up the dough bits and cut some more, from different directions, until it is in small chunks and evenly mixed with cinnamon/apples/dough etc.
Using floured hands, roughly shape into a 12-inch long x 3-inch wide log. Cut the log into 1 inch slices. With each slice, roughly pat into a disc, about the size of your palm (use some additional flour, as needed, if your dough is sticky). Place the dough discs onto a parchment lined or floured baking sheet, leaving a couple of inches of room around each. Repeat with all 12 dough pieces. Cover with plastic wrap that has been sprayed on the under-side with baking spray and set-aside to rise until doubled again, about 30-45 minutes.
Meanwhile, prepare the glaze. Add the icing sugar, corn syrup, salt and vanilla in a large bowl. Set aside. In a small saucepan, bring the water and white granulated sugar to a boil over high heat then reduce heat to medium and simmer for 1 minute. Add this mixture to the bowl with the icing sugar and whisk until all the sugar and has been incorporated and the mixture is smooth. Set aside.
When dough is about ready to cook, heat oil in a deep fryer to 370° (alternately, use a heavy deep pot and a thermometer to monitor temperature). Fry each donut until deep golden on the underside (1 1/2 - 2 minutes), flip over and fry the other side. (*time and temperature may need to vary. Watch closely the first few to determine timing. If they seem to be browning too quickly, lower temperature slightly). Remove to a cooling rack placed over a baking sheet. While still quite warm, dip into prepared glaze and return to cooling rack to cool completely.