In a large bowl, combine the flour, sugar, salt, and baking soda. Add milk and buttermilk, combine thoroughly; do not overmix. A firm dough is desired (additional flour may be added if necessary). Set aside to rise, about 15 minutes.
Transfer dough to a lightly floured board; roll to approximately 1/8-inch thick. Using a sharp knife or pastry wheel, cut into 3-inch squares.
Heat oil in a large heavy-bottomed pot over medium heat until a deep-frying thermometer registers 375 degrees F. Add the squares to the hot oil, turning over once until golden brown, 2 to 3 minutes. Using a slotted spoon, remove the beignets to a plate lined with paper towels to drain. Sprinkle with confectioners' sugar, and serve immediately.