In a bowl of an electric mixer fitted with the dough hook attachment, combine the flour, yeast, sugar, salt, baking soda, cinnamon, butter, vanilla extract, and egg. With the motor running on low, slowly drizzle in the buttermilk.
Increase the mixer speed to medium and continue mixing until the dough pulls away from the sides of the bowl and appears shiny and smooth, about 10 minutes.
Turn out the dough onto a dry surface and shape into a ball (At this point, you can place in a bowl, covered, in the fridge for up to 24 hours, if making the dough ahead of time).
Place into a lightly greased bowl, cover with plastic wrap, and let rise in a warm place (80 to 85 degrees, see notes) until doubled in size, about 1 hour.
Lightly oil a piece of parchment paper and set on a baking sheet. Punch down the dough and fold it up tightly.
On a lightly oiled surface, roll out the dough to 12 x 12-inch square (⅓ inch thickness). Using a sharp knife or bench scraper, cut the dough into 2-inch squares, making a total of 36 pieces. Place the squares on the same piece of parchment paper, cover with plastic wrap, and let rise until puffed, about 30 minutes.
While the beignets are rising, heat the oil to 375 degrees in a large dutch oven or saucepot. Carefully drop the pieces of dough, a few at a time (no more than 4), in the hot oil. Deep fry until deep golden brown on both sides, 1-2 minutes per side, flipping over with chopsticks or tongs halfway through. Be sure to keep the temperature of the oil as constant as possible.
Remove with a fine mesh strainer and place on a cooling rack set over paper towels to drain. Once slightly cooled, dust liberally with powdered sugar and serve immediately.