Put the warm milk in a large bowl and sprinkle over the yeast; let it stand for a few minutes, until it gets foamy. Add the butter, egg, sugar, half the flour and salt and stir until well combined and sticky.
Add the remaining dough, leaving about 1/2 cup to add at the end if it needs it, and stir (or knead with the dough hook attachment of your stand mixer) until you have a soft dough. Knead for 6-7 minutes, until smooth and elastic, adding the last of the flour as you need it - it should be tacky, but not too sticky.
Cover the bowl with a tea towel and let rise for an hour - it should double in size. Take out the dough and roll or pat it out on a lightly floured surface to about 3/4 inch thick. Cut into doughnuts using a doughnut cutter - or use a ring cutter or glass rim, then cut a smaller hole in the middle. (Cut them as close together as you can to avoid having to reroll the scraps - I just cook them as is, rather than try to make more doughnuts.) Cover the lot with a tea towel and let sit for 1/2-1 hour to puff up again. (They'll get puffier still as they cook.)
Heat about 1 1/2 inches of oil in a wide, heavy pot set over medium-high heat until hot, but not smoking - take a scrap of the dough and drop it in - if it sizzles and bubbles around the dough, it's ready to go.
Gently lower a few doughnuts at a time into the oil - you don't want to crowd the pot, or it will cool down the oil and the doughnuts could get oily and heavy - and cook for a minute, until golden on the bottom. Gently flip (tongs or a bamboo skewer work well for this) and cook until golden on the other side. Remove with a slotted spoon and transfer to a paper towel-lined plate or baking sheet to cool.
Set a wire rack over a baking sheet and place the doughnuts on it; whisk together the glaze ingredients and dip, dribble or brush the glaze overtop. Add sprinkles while it's still wet, before it sets.
Makes about 2 dozen small-ish doughnuts.