Stir yeast and sugar into warm water, allow to stand for 10 minutes – it should get very bubbly.
In a large mixing bowl, combine flour and salt. Pour in yeast mixture, stir well to combine.
Dump dough out onto a floured surface, knead until soft and elastic, 5-10 minutes. (OR: mix it in a stand mixer with a dough hook for 5 minutes or so!)
Once dough is fully kneaded, place in a greased bowl, cover with plastic wrap, and allow to rise for one hour, or until doubled in size.
Line a couple of baking sheets with parchment paper, set aside.
Once dough has doubled in size, roll dough out on a floured surface. Aim to make it a large rectangle, say 12 x 18" or so.
Use a pastry brush to spread melted butter all over the dough – you might not use it all, that's ok. Mix together the sugar and cinnamon, sprinkle evenly over the melted butter.
Starting with one of the longer edges, tightly roll the dough up.
Use a sharp knife to trim the ends so that they are flat, then slice the roll into 12 even rounds. Slightly flatten each roll between your hands, stretching out to an oblong shape and ensuring end is sealed to rest of dough.
NOTE: I flatten mine a fair amount more than is traditional, as I am terrible at deep frying anything thick.
Carefully place each roll onto the baking sheets, spacing them evenly and leaving room for them to rise.
Cover pans with plastic wrap, allow to rise one more time – about 45 minutes.
While waiting for the Persians to rise, start heating your frying oil to 350 F (180 C)
In small batches, deep fry doughnuts for a minute or two - until golden brown on the under side - before carefully flipping and frying for another 1-2 minutes on the second side.
Use a slotted metal spoon to transfer doughnuts to paper towel lined baking sheets or plates.
Allow oil to come back up to temperature between batches.
Allow doughnuts to cool completely.