For the doughnuts:
In a large bowl, mix together all the flour, baking powder, salt, and nutmeg.
In a large bowl, use an electric mixer fitted with the paddle attachment to beat the egg and yolk with the sugar until thick and pale, about 5 minutes. Beat in the vanilla extract. On low speed, alternately add the flour mixture with the sour cream and melted butter, starting and ending with the flour. The dough will be soft. Cover with plastic wrap and let sit at room temperature for 30 minutes, or until the dough is firm enough to handle.
Make ahead: Store the covered dough in the fridge for up to 1 day. Bring to room temperature before using.
Turn the dough out onto a floured work surface. Roll the dough out into 1/4-inch thickness and cut with a doughnut cutter or with a round biscuit cutter. I prefer doughnuts without holes so more M&Ms fit on top. Place the doughnuts on a lightly floured parchment lined baking sheet. Let the doughnuts sit at room temperature, covered, while you heat the oil.
Heat the oil in a large, deep, and heavy pan to 350°F. Place three to four doughnuts in the oil at a time and fry until golden brown, about 1 minute or less per side (30 seconds for the holes). Do not overcook the doughnuts. Drain on a paper towel-lined plate.
For the glaze:
In a medium saucepan over medium heat, combine the butter, milk, corn syrup, and vanilla and heat until the butter melts. Decrease the heat to low and add the chocolate, whisking until melted. Turn off the heat and add the powdered sugar, whisking until smooth. Immediately dip the doughnuts into the glaze. If the glaze begins to set return the saucepan to low heat and stir until liquid again. Immediately after dipping, sprinkle the doughnuts with M&Ms, pressing slightly to adhere.
Let the glaze set on the doughnuts for 30 minutes before serving.