First make the doughnuts by sifting together the flour, baking soda, cinnamon and salt into a bowl. Set aside.
In a separate bowl, whisk together the sugar, sour cream, egg and butter.
Add the flour mixture in batches to the sour cream mixture, folding it in with a spatula until fully incorporated into a smooth dough. Cover the bowl with plastic wrap and refrigerate the batter for 20 minutes.
Roll the dough out onto a lightly floured surface until it is approximately ½-inch thick.
Use a cookie cutter to cut out doughnuts that are 2½ inches in diameter. Re-roll scrap dough until all doughnuts have been cut out. Optional: Reserve the centers of each doughnut to make doughnut holes.
Prior to frying the doughnuts, prepare the sugar glaze by combining the sifted sugar, whole milk and vanilla extract in a small bowl.
When ready to fry doughnuts, heat at least 2 inches of vegetable oil in a heavy-bottomed pot until it reaches 360ºF.
Fry the doughnuts and doughnut holes in batches until they are golden brown.
Using a slotted spoon or spider, transfer the doughnuts to a cooling rack to drain. Let them cool just slightly before dipping them in the prepared glaze.