Special equipment: a doughnut cutter
For the doughnuts: Mix the all-purpose flour, cake flour, baking powder and salt with a whisk until homogenous.
In a stand mixer fitted with a paddle attachment, cream the sugar and butter until smooth, then add the eggs. Add the yogurt and mix until smooth. Slowly add the dry ingredients, then add the blueberries and juice. Do not over-mix! If the mix is too wet, add a bit of flour. Cover and let the dough rest for at least 30 minutes in the fridge.
Heat the oil until a deep-frying thermometer inserted into the oil reaches 350 degrees F (less and you will get greasy doughnuts, more you will get burnt doughnuts).
Roll out the dough on a floured surface, roughly 3/4-inch thick, then cut out the doughnuts into any shape you want. Drop the doughnuts into the hot oil a few at a time. Pay attention, this next bit is important; cook them just until they float, about 20 seconds. As soon as they float, flip them over. Once you flip them, do not mess with them!!! Just let them cook. When they are completely done on one side, flip them and cook on the other side. Again, do not mess with the doughnuts! Let them cook until done! Pull the doughnuts out and let them cool on a wire rack.
For the glaze: Mix up the confectioners' sugar, milk, salt and lemon juice to taste. When the doughnuts are cool to the touch, glaze them. Then eat and enjoy.