What is the origin of the shift of meaning of "sauté"?

sauté

VERB (sautés, sautéing, sautéed or sautéd)

[WITH OBJECT]

Fry quickly in a little hot fat:

> He sautéed the onions in olive oil.

NOUN

  1. Ballet A jump off both feet, landing in the same position.

The professional culinary use of the word is true to the image of the etymology:

1813, from French sauté, literally "jumped, bounced" (in reference to tossing continually while cooking),

past participle of sauter "to jump,"

from Latin saltare "to hop, dance," frequentative of salire "to leap" (see salient (adj.)).

As an adjective, "fried quickly," from 1869. As a verb from 1859.

The etymology does concede fry quickly as a meaning from 1869, so the culinary nuances of fry quickly seem to be under consideration. The 1851 publication, Frank Forester's Fish and Fishing of the United States and ... offers a theoretical point of departure:

Scale and well dry six Pearches, and make incisions here and there on each side of them ; then put a quarter of a pound of butter into a sauté-pan, season your fishes with pepper and salt, put them in the sauté pan and fry them gently, turning...

This is no iron-clad explanation, but if people were routinely instructed to use a sauté-pan to fry things gently in a quarter pound of butter, it would be a matter of time before this frying gently became associated with the word sauté.

In 1879, it seems confusion was again articulated in Cooley's Cyclopædia of Practical Receipts

According to the (common) mode in which all objects are cooked which are called fried, it would answer to the French word 'sauté,' or the old English term 'frizzled' but to fry any object, it should be to immersed in very hot fat, oil, or butter...

Comparing sauté to frizzle:

VERB

[NO OBJECT]

  1. Fry or grill with a sizzling noise:

    [WITH OBJECT]

1.1 Fry until crisp, shriveled, or burnt:

Since the common use has broadened so far, it might be appropriate to consider the narrower definition of sauté to be part of a professional dialect.