water
buttermilk
egg
butter
all-purpose or bread flour
sugar
salt
regular or quick-rise yeast
oil for frying
(sugar for coating the doughnuts)
(jam, chocolate hazelnut spread, and or whipped cream for filling, if desired)
Special equipment:

parchment paper, cut into about 4″ squares (smaller pieces for doughnut holes)
large pot, deep fryer or wok
a sharp doughnut cutter (a reader asked what I recommend, and I like this one as there is no guessing where the center of the doughnut is–perfect doughnuts every time).
candy/oil thermometer, not necessary, but very helpful if you’re not a deep-frying pro
skimmer

Make the Doughnut Dough
Place the water, buttermilk, beaten egg and melted butter in the bread machine pan or stand mixer bowl, then add the dry ingredients, except for the yeast.

Make a small indentation in the dry ingredients, then add the yeast.

Set the bread machine on the ‘dough’ setting.

If using a stand mixer (see directions in printable recipe below, as steps with the yeast are different), run it with the dough hook, until a soft dough is formed, then cover and set aside until at least doubled in size.

I do not specify a time on the dough rising because there are too many factors that will determine the length of time. Which yeast was used (regular or quick rise), if the yeast was fresh or old (older yeast doesn’t work as well, or sometimes not at all), and the temperature of your kitchen. In my kitchen, this process usually takes 1 to 2 hours.