Daly's Bartenders' Encyclopedia by Tim Daly (1903, p.51)

0.5 wine glasscognac base
1egg — fresh eggbase
0.5 wine glassJamaica rum base
hot milk — boiling hot milkother
nutmeg — Grate on top.other
2 spoonfulssugar sweetener
  1. Add the egg, sugar, Cognac, and Jamaica rum to a mixing glass.
  2. Fill with boiling hot milk while stirring the contents well, grate nutmeg on top, and serve.

Garnish: Nutmeg.

Daly explicitly frames this as tonic and strengthening service, which makes it a good restorative witness.

The hot milk incorporation while stirring is the central service logic.

Jack's Manual by J. A. Grohusko (1910, p.58)

1 portionbrandy — Source gives 100% brandy.base
1 tablespoonfulsugar base
1 fresh eggegg other
hot milk — Fill glass with hot milk.other
nutmeg — Grate nutmeg on top.other
  1. Shake the sugar, egg, brandy, and hot milk thoroughly with a shaker.
  2. Strain, grate nutmeg on top, and serve.

Garnish: Grated nutmeg

Ice: None

The source explicitly calls for shaking thoroughly with a shaker before straining.

See also: Egg Nogg, Hot Milk Punch