For the Dough 

1. Preheat your oven to 200 degrees Fahrenheit for rising purposes. As soon as the oven preheats, you will turn off the oven. This will provide a warm space for your dough to rise. You can also run your dryer before beginning the baking process or just find a warm spot in your house.
2. In a medium/large bowl, add warm water, warm milk, butter, egg, sugar, yeast and salt into the bowl. Stir with a wooden spoon until the mixture is smooth and combined.
3. Slowly add flour to the bowl, mixing with a wooden spoon or electric mixer. When flour is fully mixed, the mixture should be sticky and look pretty wet. These donuts are soft, light, and airy, so the dough should not be tough or dry. You should be able to touch it with floured hands without any of the dough being transferred to your hands. If it transfers to your hands, add another tablespoon or two of flour at a time until it is not longer sticky and wet.
4. If using a dough hook, knead for 6-7 minutes. If kneading by hand, turn your dough out onto a clean, floured surface. Knead until dough is stretchy, elastic, and tacky. Massage the dough with the palm of your hand, fold it, turn it, and repeat. Add flour by the teaspoon to your dough to help with the stickiness of the kneading process. The longer you knead, the easier and less sticky the process will become.
5. Transfer dough to a large, oiled bowl. Roll the dough around in the oil. Cover the bowl with cling wrap or a damp cloth before placing it in your warm oven, warm dryer, or in a warm place in your house. Let the dough rise for one hour until it has doubled in size

To Assemble

1. Once the dough has risen, push down on it to release the air. On a clean, lightly floured surface, roll the dough out to be 1/4 inch thick using a rolling pin. Use a cup, bowl, or cookie or pastry cutter to cut 3-4 inch rounds. This doesn’t need to be exact, but try to make sure all of the donuts are the same size even if they aren’t 3.5 inches. Note that the donuts will grow in size after the second rise.
2. You may need to re-roll the scraps of the dough to get a few extra donuts. Simply combine the scraps together and use the rolling pin to roll it to 1/4 inch thickness again. Only reroll the dough once. 
3. Transfer donuts with a spatula to individually greased parchment slips on a large plate or pan. This will allow them to easily remove from their surface, keeping their shape as they are being fried later on.
4. Use a coring tool, piping tip, or a sharp knife and spoon to cut about 1 inch rounds out of the center of each donut to create the hole. I used the top of a large, round piping tip to do this.
5. You can put the middle of the donuts on their own parchment slips to make donut holes.
6. Cover the donuts with a damp towel and allow them to rise in a warm place for another 30 minutes. You can re-preheat your oven if it has gotten cool.

To Fry

1. In a medium-sized pot, add oil to fill 3-4 inches. The amount you need depends on the depth of your pan. The amount of oil doesn’t matter too much, only that it isn’t too shallow in the pot. The donuts will stick to the bottom of the pot if there isn’t enough oil.
2. On medium heat, bring the oil to 315 degrees Fahrenheit. Once it has reached this temperature, lower the heat slightly to ensure it doesn’t heat up too much more.
3. Slide as many donuts that can fit in the pot into the oil. Allow the donuts to cook for about 1 minute on each side, or until the donuts reach a light golden brown color. Use a spatula to flip them and remove them.
4. Once you remove the donuts from the oil, place them on a paper-towel- lined plate to soak up some of the excess oil.

For the Frosting

1. After all donuts have been cooked, begin making the frosting. If you would like to make both flavors, I’d recommend halving each recipe. Simply whisk the ingredients together in a medium-sized bowl. Frostings should not be too runny or thin. They should be thick enough that they don’t look wet or watery but thin enough that they fall off of a spoon when the spoon is lifted out of the bowl. If the frosting is too thin, add confectioners’ sugar by the Tablespoon. If the frosting is too thick, add milk by the teaspoon. Thick frosting will stick to the donut and will not go on smoothy, and thin frosting will spread and turn into more of a glaze.
2. Once the donuts have slightly cooled, dip the top of the donut into the icing. Twist the top of the donut around to better coat it. If you are using sprinkles, dip the donuts into a bowl of the sprinkles if you want the donuts to be fully coated, or simply use your hands or a spoon to drizzle them on top if you want a lighter topping of sprinkles. You’ll want to make sure you immediately add sprinkles or toppings after dipping the donuts because the frosting will set.
3. Serve as soon as possible for best texture. Cover for up to 3 days in an airtight container at room temperature.