In a bowl add the flour, sugar, salt and nutmeg. Mix
In another microwave-safe bowl, add the butter, and melt it. Add the egg, the milk, the instant yeast and mix.
Pour this into the initial flour mixture and mix with a spatula. The dough will be quite sticky. Although it is tempting, do not add flour. This sticky texture will give us very tender doughnuts.
On a surface, add a pinch of oil and pour the dough. Knead. If you have a Kitchen Aid, you can use it on medium speed for 5 minutes. Otherwise, knead the dough by hand for about 15 minutes until the dough is elastic and smooth.
Put the dough in a bowl and cover it with cling film. Leave it to rest for an hour and a half or until it has doubled in size.
After this time, flour the surface. Pour the dough, flour the top a little to prevent it from sticking, and begin to roll it out until it is half an inch thick.
Cut the doughnuts. You can use a doughnut cutter or like me, two cookie cutters. One about 3 inches and the other about 1 inch.
Transfer the doughnuts to a baking paper to prevent them from sticking and repeat with the remaining dough. Roll out the dough and cut it out. When there is not enough dough left to make more doughnuts, do not throw it away, you can make doughnuts holes.
In a frying pan, add the oil and heat it to 160ºC, controlling the temperature well, and fry until golden brown. Remove.
In a bowl add the icing sugar, the butter, the milk and the vanilla extract. Mix together.
Add the donuts, one at a time, coat them well. Transfer to a rack and let them to rest until the glaze sets.