Put the water in a large bowl and sprinkle over the yeast. Let it sit for a few minutes to dissolve. (If you’re worried that it might be old or inactive, wait a few more minutes and watch for it to foam—if it doesn’t, you may need fresh yeast. If you're using instant or quick yeast, no need to even dissolve it-you can just add it to the bowl along with the liquid, flour etc and it will do its thing.) Add the flour, milk, butter, egg, sugar and salt, and stir, then knead until you have a soft, very tacky dough. Cover and set aside for an hour or two, or until about doubled in size.
Turn the dough out onto a lightly floured surface and pat with floured hands until it’s about 1/2 inch thick. Cut out as many rounds as possible, as closely together as you can to avoid scraps, using the rim of a shot glass or open end of a tomato paste can. Poke a hole in each with a chopstick or your finger, stretching it out a bit as it will puff up as it cooks—make the hole bigger than you want it to be once it’s cooked. If you like, cover with a kitchen towel and let them rise for another 20-30 minutes (this isn’t necessary, but will produce lighter doughnuts).
Heat about an inch and a half of oil in a deep, heavy saucepan (I use an enamel-coated cast iron Dutch oven) until it’s hot but not smoking—if you have a thermometer, aim for about 350F, and if you dip a scrap of bread or bit of dough in, it should start to sizzle. Cook the doughnuts about 6 at a time, depending on the size of your pot (you just don’t want to crowd it too much), turning occasionally with tongs or a slotted spoon, until puffed and golden brown, about 2 minutes per batch. Reroll the scraps once, or just cook them as scraps—these pieces tend to be our favourites.
Transfer to a paper towel-lined plate for a few minutes, while you put some sugar into a shallow dish and stir in a generous shake of cinnamon. Toss the doughnuts in the cinnamon-sugar while still warm. Makes lots.