Add dough ingredients to bread machine in order recommended by your manufacturer. Set to dough cycle.
If you're saving some dough for later, divide dough in half and place half in the refrigerator, wrapped in cling wrap.
Roll out remaining dough to about ½" thickness.
Preheat oven to 200°, if possible, turn off at around 150°. Prepare some boiling water (either on the stove or in the microwave) in an oven save dish.
Using a doughnut cutter, or round cookie cutters, cut the desired doughnut shape. If you are filling your doughnuts do not cut an inner circle. Continue to combine and re-roll dough until you have the optimum (desired) number of doughnuts, and doughnut holes (from inner doughnut circle).
Place the donuts and donut holes on parchment paper, then placed on a cookie sheet. Place the cookie sheet in the oven with the boiled water and close the oven. NOTE: to make frying easier, precut the parchment paper to individually sized donut pieces. NOTE: the goal of the oven is to create a warm, humid environment for the donuts to rise. You do not want it to be too hot so that it pre-bakes, or too dry so that a skin forms on the donuts. Allow the donuts to rise until doubled (or more) in size. The more they rise, the fluffier the doughnuts.
While the donuts are rising, preheat your oil. The ideal temperature is around 350°, but you can go by eye, just making sure that the doughnuts are more of a golden color than browned.
Add the doughnuts to the oil, frying on each side until just golden brown, then placing on a wire rack (or paper towel lined plate) to cool.
While the doughnuts are cooling, prepare your glaze. Simply mix all of the glaze ingredients for your flavor of choice with a whisk until smooth. Note that the glaze will harden, so do not prepare too long in advance. If making more than one or two flavors at a time I suggest cutting the recipes in half.