The phrase the cold shoulder denotes a show of intentional and marked coldness or of studied indifference.

Because the two earliest instances of this phrase recorded in the Oxford English Dictionary (2^nd edition – 1989) are from the Scottish novelist and poet Walter Scott (1771-1832) and do not refer to food, Robert Allen writes in Dictionary of English Phrases (2008) that this author probably invented the expression or picked it up from local usage, and did not allude to a cold shoulder of mutton or veal offered as a dish to an unwelcome visitor but to the physical action of turning or hunching the shoulder as a gesture of unfriendliness or indifference.

The first quotation is from The Antiquary (1816):

“O, doubly did she hate Eveline Neville when she perceived that there was a growing kindness atween you and that unfortunate young leddy! Ye may mind that the Countess’s dislike did na gang farther at first than just shewing o’ the cauld shouther—at least it wasna seen farther; but at the lang run it brak out into such downright violence that Miss Neville was even fain to seek refuge at Knockwinnock castle with Sir Arthur’s leddy.”

Scott probably thought that the phrase would be unfamiliar to the readers, as he included it in a glossary at the end of the book:

Shouther. Shoulder. Show the cauld shouther, Appear cold and reserved.

The second quotation is from St. Ronan’s Well (1823):

“I must tip him the cold shoulder, or he will be pestering me eternally.”

The verbs show and tip in these passages do suggest a physical gesture rather than an offer of food.

I have found an earlier figurative use of cold shoulder in a humoristic article published in The Chester Chronicle (Cheshire) of 22^nd July 1808 (eight years before The Antiquary was published); it seems to indicate that the phrase refers to a dish of cold shoulder of mutton or veal offered to an unwanted guest:

‘Fashions.’—Since the commencement of the present intolerably ‘hot weather,’ ‘warmth’ is all the rage; and yet, strange to say, a fine Lady is ready to faint at the idea of a ‘fleecy hosiery’ petticoat. The ‘coldest’ friends cannot meet without a ‘warm’ embrace, and even old maids, who have numbered 60 winters, confess themselves in a ‘melting’ mood. The ‘Restaurateur’ says he has ‘hot’ work of it in his ‘ice’ house, and the glass-blower, just released from his furnace, complains of the ‘sultry’ air out of doors. The public plume themselves upon their ‘ardour’ in the Spanish cause, and reprobate all that are ‘lukewarm,’ as ‘cold-blooded’ traitors. The rich miser looks ‘coldly’ upon his poor friend, and the inhospitable landlord treats his uninvited visitor to a ‘cold’ shoulder.

But, as the author of this article plays, in the context of a heatwave, on the differences between the figurative and the literal meanings of words relating to coldness and warmth, it is not certain that cold shoulder as a phrase originally alluded to a dish. What is certain on the other hand is that Walter Scott did not coin or popularise the phrase.

In The Mirror of Fashion, published in The Morning Chronicle (London) of 4^th October 1823, the year St. Ronan’s Well was published, I have found a puzzling instance of cold shoulder: it seems to be some allusion which is now unfortunately impossible to understand:

The Duke and Duchess of Bedford and family left town yesterday morning for Brighton.
The Earl and Countess of Darnley and family have arrived at Ramsgate, from their seat, Sandgate, Kent.
[…]
The guests at the Windsor dinner given in the open air, in addition to the fare on the table, were each of them treated with a ‘cold shoulder.’
‘Ambition’ appears to be on the vane among our theatrical females. The Green-rooms now afford only one aspirant to a ‘Coronet.’