Whisk flour, cocoa, sugar, baking powder, baking soda, salt, and nutmeg in a large bowl.
In another bowl, whisk milk, sour cream, egg, vanilla, and melted butter until smooth.
Stir wet ingredients into dry just until combined; the batter will be thick. Rest 5 minutes.
Heat 2–3 inches of oil to 350°F in a deep pot, keeping a thermometer clipped to the side.
Scoop heaping tablespoons of batter and gently drop into the oil. Fry 2–3 minutes, turning for even browning.
Transfer to a rack set over a sheet pan to drain. Keep oil near 350°F between batches.
Whisk powdered sugar, milk, vanilla, salt, and corn syrup until smooth and pourable.
Dip warm donut holes into the glaze, letting excess drip off. Set on the rack to firm up.