A deep-fryer isn’t necessary, any heavy bottomed pot will do.
Fill the pot at least 3 inches deep with oil so that the doughnuts have enough room to float.
Use a thermometer (I use a candy/deep-fry combo thermometer like this one) and fry the doughnuts at 375 degrees F. If you try to fry doughnuts in oil that is too cool, it’ll take longer and your doughnuts will soak up more oil. If the oil is too hot, you risk burning the doughnuts.
If you don’t have a thermometer, test the temperature with a single doughnut hole. If it immediately sizzles and rises to the surface, the oil is ready.
Adding the doughnuts to the hot oil will cause the oil temperature to drop. Fry in batches and adjust heat as needed to maintain 375 degrees while you fry the doughnuts.
Protect yourself! Wear an apron to protect your clothing from any splatters, and use a slotted spoon to gently ease the doughnuts into the hot oil, flip them over, and remove them.
When you are done, turn off the heat and let the oil cool to room temperature completely. Pour through a funnel into a container with a secure lid and discard (or save to use for frying later). Do not pour oil down the drain.