1. Mix 3 T of the warm milk and yeast together.
2. Set yeast mixture aside for 5 minutes.
3. Mix the remaining 1/2 C of warm milk, butter, and sugar.
4. Add yeast to above mixture.
5. Mix in the egg, flour, nutmeg, and salt until the flour is incorporated.
6. Turn out the dough onto a floured work surface and knead until smooth and elastic.
7. Shape it into a smooth ball.
8. Transfer the dough to an oiled bowl and cover with moist towel.
9. Let the dough rise until its doubled in size.
10. Preheat oven to 375F.
11. Punch down the dough and roll it out 1/2-inch thick on a floured work surface.
12. Cut out 2 - 3 inch diameter circles.
13. Transfer the circles to a parchment-lined baking sheet.
14. Cut out 1 inch diameter holes in the centers.
15. Cover the baking sheet with a clean cloth and let the doughnuts rise for another 45 minutes.
16. Bake in a 375F oven until the bottoms are just golden, 8 to 10 minutes.
17. While the doughnuts are baking, melt the 1/4 C of butter.
18. Combine sugars and cinnamon in separate bowl.
19. Remove the doughnuts from the oven and let cool for a minute or two.
20. Dip each donut in the melted butter and a quick toss in the sugars.

It is better to underbake then overbake here and so pull the donuts early if in doubt.