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The colloquial phrase milk moustache (also milk mustache, moustache of milk and mustache of milk) designates a white residue, resembling a growth of hair above the upper lip, left after drinking milk.

This phrase occurs, for example, in the following from the column The Spectator, by Carla Carlisle, published in Country Life (London, England) of Thursday 8^th May 1997 [page 104, column 4]—Joanna Lumley (born 1946) is a British actress best known for her role in the BBC sitcom Absolutely Fabulous:

‘MILK. Where’s your moustache?’ is the theme of the campaign paid for by the (American) Fluid Milk Promotion Board.
[…]
[…] Even if I can get my own family to sport the milk moustache, there is a whole world out there that needs to get drinking. Meanwhile, I’m sending the British Milk Development Council a few hints on promoting milk in this country, beginning with a poster of Joanna Lumley sporting a Tipp-Ex milk moustache.

These are, in chronological order, the earliest occurrences that I have found of the phrase milk moustache (also milk mustache, moustache of milk and mustache of milk):

1-: From Poor Miss Finch. A Novel (London: Richard Bentley and Son, 1872), by the British author Wilkie Collins (1824-1889) [volume 1, chapter 10: First appearance of Jicks, page 115]—Jicks, “a chubby female child, who could not possibly have been more than three years old”, is the one who begins the sentences with “Jicks will”:

“Jicks will have something to drink.”
[…] Oscar ran into the kitchen for some milk. […]
[…]
Oscar returned with the milk in a mug. The child—insisting on taking the mug into her own hands—steadily emptied it to the last drop—recovered her breath with a gasp—looked at me with a white moustache of milk on her upper lip—and announced the conclusion of her visit, in these terms:
“Jicks will get down again.”

2-: From Table Manners Should Be Taught Children Very Early: Correct Ways of Holding Fork and Spoon Must Be Learned in Early Infancy—Proper Way of Holding Glass and Drinking Essential, by Rosanna Schuyler, published in The Washington Times (Washington, District of Columbia, USA) of Tuesday 7^th May 1907 [page 7, column 3]:

Comparatively few children are taught how to drink in a well-mannered way. Instead of the cup being allowed to go into the mouth to the corners, the little one should learn to take small mouthfuls of the liquid, and the lips will be less covered with it when the cup is taken down. “Milk moustaches” should be as reprehensible for children as for grown persons.

3-: From The True Story of Kitty White, a short story by Ethel Bowen White, published in the Raleigh Christian Advocate (Raleigh, North Carolina, USA) of Thursday 12^th June 1919 [page 10, column 2]—reprinted from The Christian Register:
—Note: Kitty is a cat:

The promised milk tasted most refreshing. Kitty White drank hers from a nice blue-and-white Japanese saucer, while her master drank his from a blue-and-white Japanese cup. When Frankie saw Kitty forget to wipe her milk mustache away, he remembered first to wipe his own mouth with a fresh handkerchief, then he wiped Kitty’s with one of the feathers left in the basket.

4-: From Correct Ways to Do The Little Things, published in the Pasadena Star-News (Pasadena, California, USA) of Saturday 15^th October 1921 [page 8, column 4]:

Artificial straws, each encased in its sanitary wrapper are now generally served with all sorts of beverages, including milk, and the different milk combinations concocted at soda fountains and in cafes. There are hygienic reasons for the use of these cleanly tubes, each of which is discarded after use, but there are other reasons, too, and these are concerned with seemly table manners.
For instance, if a glass of milk is ordered, or a milk-shake, or frosted chocolate, or hot chocolate with whipped cream, or ice cream soda, and no straws are served with the order, it is difficult to drink the liquid without getting a “moustache” of milk or cream or milky chocolate around the lips. This disagreeable looking milksop effect may be avoided by taking the beverage through a straw. It must be done without making any whistling or gurgling noise during the tube-drinking, this can be practiced at home.

5-: From a letter to the Editor, about the No Fence Law, by “a man on Tennessee river”, published in The Dover Courier (Dover, Tennessee, USA) of Friday 31^st March 1922 [page 8, column 2]:

Little children here that now have plenty of milk and butter to stand back in the corner and make milk mustache and if this law should pass a few of them would not know how milk and butter tastes unless they went to some of their more fortunate neighbors.

6-: From Lonely Women, a short story by the U.S. author Jay Gelzer (1889-1964), published in The Kansas City Star (Kansas City, Missouri, USA) of Sunday 6^th April 1924 [Society section, page 1E, column 6]:

The sick man’s eyes were open, and, as Clare had said, they were of an intense blue, childishly appealing. He had a pleasant, friendly smile, which he produced for Blue Allard.
“Thought you were a man, when you first come in that door,” he said, smiling. “Mighty sorry I’ve had to put you to all this trouble.”
“No,” denied Blue Allard, with an odd breathlessness, “I’m not a man. I’m a woman.”
“You’re the boss around here, ain’t you?”
“I’m the boss,” nodded Blue Allard.
She sat down beside him in a silence which lengthened until the sick man stirred restlessly.
“Thought I smelled doughnuts,” he hazarded weakly.
“Clare!” called Blue Allard grimly.
Clare came at once, bearing a plate of the despised doughnuts and a glass of milk. He ate, smiling at intervals at Clare, a mustache of milk lingering upon his upper lip, which Blue Allard wanted absurdly to wipe away.