Stir the lemon juice into the milk and set aside until milk thickens. In a medium bowl, mix together the shortening and ¼ cup white sugar and ¼ cup brown sugar until well incorporated.
Beat in the egg and vanilla until all of the lumps are gone and mixture is smooth.
In a small bowl sift together the flour, baking soda, cinnamon, nutmeg, and salt. Alternate stirring in flour and milk into the egg mixture until a soft dough forms.
Dust a work surface with flour and briefly knead the dough. Pat out the dough into a disc and roll out to about 1/3-inch thickness.
Using a donut cutter, a biscuit cutter, or a glass and cut out as many donuts as you can. Re-roll out dough scraps to use up all of the dough. If you don’t have a donut cutter, use your fingers to poke a hole in the middle.
Lay out doughnuts on a wax paper lined tray and let stand for 10 minutes. Heat the oil to 375 degrees. While waiting for the oil, make the sugar coating.
In a bowl , mix together the remaining sugar and cinnamon, set aside.
To fry donuts, gently drop 3-4 donuts in the hot oil and cook for 1-2 minutes on each side, flipping once (the handles of wooden spoons or wooden chopsticks are the best tools here). Remove from the oil and drain on a wire rack. Continue with the rest.
To coat the donuts, place the still warm donuts, one at a time, into the sugar mixture and jiggle a little bit so that you get a nice sugary coating. Set back on the wire rack and continue with the remaining donuts.