Make the Doughnuts: In a small saucepan, warm the milk slightly with 1 teaspoon sugar. Add the yeast and let sit for 5 minutes to proof.
In a small bowl, beat the egg with the water.
In a standing mixer with a paddle attachment mix the flour, remaining 1/3 cup sugar, salt, and nutmeg. Add the proofed yeast mixture, beaten egg mixture, and melted butter. Remove the paddle, switch to a dough hook, and mix until well-blended and kneaded, about 15 minutes. Transfer the dough to a large greased mixing bowl. Cover with plastic wrap and chill overnight.
On a floured surface, roll the dough out to about 1/4-inch thick and transfer to a sheet pan. Cover with plastic wrap, and chill for 30 minutes, allowing the dough to rest.
Cut dough into 1 1/2-inch squares. Line a sheet pan with a buttered sheet of parchment paper. Arrange the dough squares 1/2-inch apart on the sheet pan and cover with another sheet of buttered parchment. Place in a warm place to rise about 30 minutes.
Make the Coating: In a small bowl, combine the sugar and cinnamon to coat doughnuts.
Heat the oil in a large pot to 375 degrees F and fry the doughnuts for 1 to 2 minutes on each side. While doughnuts are still warm, toss them in the cinnamon sugar, to coat.