In a small saucepan, heat the milk over medium heat until small bubbles start to form around the edges. Pour into a large bowl (one that goes in a stand mixer, if you have one) and stir in the buttermilk. Stir in the yeast and sugar and set aside for 5 minutes, until the yeast starts to bubble. (If it does nothing, it may be inactive - you’ll need fresh yeast.)

Add 2 cups of the flour, the baking soda and salt and mix until well blended. Add more flour, 1/2 cup at a time, and continue mixing for a few more minutes, until the dough starts to come together but is still very sticky - you wouldn't be able to knead it if you wanted to. Cover the bowl with a towel and set aside in a warm place for an hour, until doubled in bulk. At that point it shouldn't be sticky anymore.

On a lightly floured surface, roll the dough out 1/2-inch thick. (You may need to dust the surface of the dough with flour as well to keep it from sticking.) With a sharp knife, dough scraper or pizza wheel, cut the dough into 2-inch squares. (They will puff up as they cook - you don’t want them too big, or they won’t cook all the way through.) If you want to make them ahead, place them on a parchment-lined sheet, cover and keep them in the fridge for up to 8 hours before frying.)

In a large, heavy pot, heat a couple inches of canola oil until it’s hot but not smoking - it should read 375°F on a thermometer. Cook the beignets a few at a time - without crowding the pot, or the oil will cool down and the beignets will wind up greasy - until golden on each side. It should take about 4 minutes - turn them with tongs as you need to, then remove with a slotted spoon and set on a paper towel-lined plate to cool. Serve warm, doused generously in icing sugar.