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The expression hen’s milk designates a drink similar to eggnog, made from eggs or egg yolks whisked into water or milk, typically sweetened with sugar, flavoured with rum, brandy, orange flower water, etc., and served warm.

Now usually associated with Christmas, hen’s milk was originally popular as a restorative and as a remedy for colds.

The expression hen’s milk is a loan translation from French lait de poule.—However, cf., below, note 1.

The earliest occurrence that I have found of the English expression hen’s milk designating a drink made from an egg yolk whisked into warm water is from The French Family Cook: Being a complete System of French Cookery. Adapted to the Tables not only of the Opulent, but of Persons of moderate Fortune and Condition (London: Printed for J. Bell, 1793) [page 227]:

Different Ways to dress Eggs.

Excepting meat, nothing furnishes a greater variety in the kitchen than eggs; but before I proceed to the various ways of dressing them, I shall speak of their utility. The yolks of new-laid eggs beat up in warm water, is called hen’s milk, and, taken going to bed, is good for a cold.

The French Family Cook was translated from La Cuisinière bourgeoise, suivie de l’Office à l’usage de tous ceux qui se mêlent de dépenses de Maisons (Paris: Chez Guillyn, 1746), by the French author of cookery books Joseph Menon (1700?-1771).—Cf. also, below, note 2.

In La Cuisinière bourgeoise, the corresponding passage is as follows [Chapter 12: Des Œufs, page 256]:

Après la viande rien ne fournit une plus grande diversité en cuisine que les œufs […].
[…]
Je dirai ici quelque chose de leur proprieté avant que d’en venir aux differentes façons de les accommoder.
Le jaune d’œuf frais délayé dans de l’eau chaude avec un peu de sucre, le boire en se couchant est bon pour les personnes enrhumées, c’est ce que l’on appelle lait de poule.

Notes:

1 I have, however, found an isolated early occurrence of English hen’s milk, translating the Italian expression latte de gallina and designating “some excellent dish”. This early occurrence is as follows, from A New Plain and Useful Introduction to the Italian. Compiled from the best Grammarians, who have wrote in the Tuscan Language. Together with a Choice Collection of Italian Idioms. Collected from the most noted Authors, with the proper English adapted (London: Printed for J. Wilcox, 1739), by John Kelly (1680?-1751) [A Collection of Idioms, Phrases, Cant-words, &c.; s.v. L, page 348]:

Latte di gallina. Hen’s milk; by this is understood some excellent dish, whatever can be imagin’d the most excellent.

2 In the sense of a drink made from an egg yolk whisked into warm water, the French expression lait de poule then occurred in the following two texts:

1-: From Dictionnaire de la langue françoise, ancienne et moderne, de Pierre Richelet, nouvelle édition, augmentée d’un très-grand nombre d’articles (Lyon: Chez Pierre Bruyset-Ponthus, 1759) [Vol. 2: E—O; s.v. lait, page 491, column 1]:

Lait de poule. C’est un jaune d’œuf de poule battu dans de l’eau, que l’on prend fort chaud. Son usage est bon pour la poitrine, apaisse la toux, &c.
     translation:
Hen’s milk. It is a hen’s egg yolk whisked in water, that is taken very warm. Its use is good for the chest, soothes cough, &c.

2-: From Précis de la médecine pratique, contenant l’histoire des maladies, dans un ordre tiré de leur siège ; avec des observations & remarques critiques sur les points les plus intéressans (Paris: Chez Vincent, 1759), by the Fench physician Joseph Lieutaud (1703-1780) [Book 1: Les Maladies internes; Section 3: Maladies de la Poitrine; Tussis, page 245]:

Dans la toux & la fiévre catarrhale simples, […] les remedes dont on use le plus familiérement […] sont la réglisse, […] le lait de poule, les tablettes de guimauve, de sucre d’orge […].
     translation:
In simple cough and catarrhal fever, […] the remedies that are the most commonly used […] are liquorice, […] hen’s milk, marsh-mallow tablets, barley-sugar tablets […].