Heat the milk to 110 degrees F. Place in a cup and stir in 1 1/2 teaspoons of the sugar and the vanilla bean seeds and pod. Sprinkle the yeast over the top and let sit for 10 minutes. The yeast will begin to bubble. Remove the bean pod, rinse, and set aside for another use.
In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, the remaining 1/3 cup sugar, the salt and the nutmeg. On low speed, stir just to mix. Add the yeast mixture to the dry ingredients on low speed, then add the egg, water and the melted butter.
Change to a dough hook and mix on medium-high speed until the dough pulls away from the sides of the bowl, 15 to 18 minutes. Put the dough in a greased bowl, cover with plastic wrap and refrigerate overnight (this is the first proof).
On a floured work surface, roll the dough out into a rectangle 1/4 inch thick. Transfer to a sheet pan and refrigerate for 30 minutes to rest the dough.
Cut out thirty 4-by-4-inch squares of parchment paper. Lightly oil each one or spray with pan spray. Have several baking sheets ready.
Remove the dough from the refrigerator and place on a work surface. Cut out doughnuts with a round cutter (or cut squares for a different look. If you're going for the traditional doughnut shape, use a smaller cutter to make a hole in the center of each round. (Do not cut out holes if you are going to fill the doughnuts. You may decide to cut the doughnuts really large; that's also fine.)
Once each doughnut is cut, set it on an oiled square of parchment. (Trust me, this will make your doughnut experience much better and more efficient.) Place the doughnuts on baking sheets. Cover the sheets lightly with an oiled piece of parchment. Allow to proof about 45 minutes in a warm, draft-free area. They are ready when they are almost doubled in size.
Heat 3 to 4 inches of oil in a straight-sided heavy saucepan. (I like rice bran oil or grapeseed oil, but vegetable oil or canola are also good.)
Clip a deep-fat thermometer to the side of the pan and heat the oil to 350 degrees F. Place the cinnamon sugar in a bowl to dredge the warm doughnuts.
Lift a doughnut on the square of paper and place it just over the oil; this will allow the doughnut to slip right into the oil without deforming. Use tongs to nudge it off the paper and into the oil if necessary; if you must, the paper and doughnut can even go into the oil together and you can immediately fish the paper out with the tongs. You can fry 2 or more doughnuts at a time, just don't over crowd the pan. I like flipping the doughnuts quickly and often to help them keep their shape. I use two chopsticks, placing them on the edge of the doughnut and using a quick motion to flip them. It will take about 1 minute on each side until they're golden brown. (Keep an eye on the heat, especially if frying several doughnuts at once.) When browned, carefully lift the doughnuts out of the oil, drain briefly, and toss in the cinnamon sugar. Place on a rack to cool. Repeat until all the doughnuts are cooked.
These are best eaten within a few hours of frying. They can be reheated in a low-temperature oven to give them that just-fried texture.
If you'd like to use a filling such as a jam, jelly, pudding, curd or anything that you can think of, it should be placed in a piping bag with a special doughnut filling tip or plain round tip. Find a spot on an edge of a doughnut, insert the tip, and squeeze the filling into the doughnut. When you feel the doughnut become heavy and the filling starts to come out, you know that doughnut is filled to the max. Enjoy!