Part of the problem may be that this usage pertains to the adjective subject rather than the verb.
One definition at dictionary.com for the adjective is:
19. open or exposed (usually followed by to ): subject to ridicule.
"These computers are subject to change" means the computers are open to change.
Another part of the problem with this usage may be the alternate definitions:
Some other definitions at dictionary.com for the adjective are:
20. being dependent or conditional upon something (usually followed by to ): His consent is subject to your approval.
21. being under the necessity of undergoing something (usually followed by to ): All beings are subject to death.
The range of definitions from "open to exposed" to "being dependent or conditional upon" to "being under the necessity of undergoing" almost seems designed to confuse.
To follow up on FumbleFingers's OED quote, I note that the "to look over or through in a casual or cursory manner" definition is a relatively recent addition to the Merriam-Webster's entry for peruse. That definition first appears (in the Collegiate series) in the Tenth Collegiate Dictionary (1993). The Ninth Collegiate Dictionary (1983) has a substantially shorter entry for peruse:
1: to examine or consider with attention and in detail: STUDY 2: READ
Perhaps the most ambiguous definition of peruse in the Tenth Collegiate Dictionary is the revised form of the older definition 2 ("READ"):
2: READ; esp., to read over in an attentive or leisurely manner
This definition remains unchanged in the Eleventh Collegiate Dictionary (2003), still the most recent edition in the Collegiate Series. To me, "an attentive or leisurely manner" is a bit like "a hard-working or indolent manner"—it covers a lot of ground in two places with very little overlap between them.
Because the word can mean quite different things nowadays, I wouldn't assign a more specific meaning than "read" to any contemporary occurrence of it unless I had contextual clues to help me interpret the speaker's (or writer's) intent.
FOLLOW-UP: Various grammar and usage commentators have addressed the proper usage of peruse. For example, Eric Partridge, Usage and Abusage, Fifth Edition (1957) offers this entry:
peruse is not synonymous with 'to read', for it means to read thoroughly, read carefully, from beginning to end. One peruses a contract, one reads an (ordinary) advertisement—that is, if one does not merely glance at it.
Bryan A. Garner, Modern American Usage (2003) expresses a similar view:
peruse (= to read with great care) is pompous and stilted in business correspondence. That is, the word shouldn't be used merely as a fancy substitute for read. ... Some writers misuse the word as if it meant "to read quickly" or "scan" [examples omitted]. That slipshod extension has become common enough to be listed in some dictionaries. But since it's the opposite of the word's traditional meaning, that usage is best shunned.
Of course, both Partridge and Garner focus on how peruse should be used, not on how people actually use it. For a descriptivist view—and a critique of prescriptivist hostility toward using peruse to mean simply "read"—we can consult Webster's Dictionary of English Usage (1989). Its discussion runs for a page and a half.
Briefly, WDEU argues that peruse arose as "a literary word" that, in poetry, served as "a useful alternative to the monosyllabic read." WDEU traces hostility to the use of peruse to mean simply "read" to Frank Vizetelly, A Desk-Book of Errors in English (1906), which made the following assertion:
peruse should not be used when the simple read is meant. The former implies to read with care and attention and is almost synonymous with scan, which is to examine with critical care and in detail. A person is more apt to read than to scan or peruse the Bible.
Incidentally, scan has long since completed the bipolar metamorphosis that some people now attribute to peruse. As Garner notes, "scan is ambiguous: it may mean either (1) 'to examine carefully, scrutinize' or (2) 'to skim through, look at hurriedly.' In [American English], as it happens, sense 2 now vastly predominates—a tendency bolstered by the ubiquitous electronic scanner, which contribute to the idea of haste."
According to WDEU, "it appears that this notion of the correct use of peruse was Vizetelly's own invention. It was certainly born in disregard of dictionary definitions of the word and in apparent ignorance of the literary traditions on which those dictionary definitions were based."
WDEU then cites 21 examples (ranging in publication date from 1594 to 1968) involving peruse—some with a narrowing adverb such as thoroughly, diligently, or attentively, or (contrariwise) negligently or idly, and others with no adverbial modifiers (such as "I perused a number of public notices attached to the wall," from a 1939 book by Flann O'Brien). Then it makes its central argument:
You may have noticed by now that the plain word read can readily be substituted in any of these examples, even where the idea of "read through or over" is pretty obvious.
...
In conclusion we recommend that you reread the examples and see for yourself in how many Samuel Johnson's simple "read" definition would work perfectly well. There are likely to be only a few in which adding the adverbs used by later dictionary definers will enhance anyone's understanding of the passage.
Under the circumstances, I don't think that the Tenth Collegiate's 1993 expansion of the second definition of peruse four years after WDEU appeared from "READ" to "READ; esp., to read over in an attentive or leisurely manner" represents a disavowal by Merriam-Webster's of the generic sense "read" in favor of something along the lines of "either read in an attentive manner or read in a leisurely manner, but not simply read." Rather, I think it reflects Merriam-Webster's desire to call out the two most common narrower senses that peruse-as-"read" takes, while upholding the continued validity of the root meaning "read."
In any event, Merriam-Webster's republished WDEU as Merriam-Webster's Dictionary of English Usage in 1994 (one year after the Tenth Collegiate appeared) with its lengthy discussion of peruse unchanged.
Best Answer
This phrase is not common, and appears to just be a diversion into cooking terms to create an analogy.
Sautéing is as you defined it. When you sauté olive oil, garlic, and onion together when starting to make an Italian pasta sauce, the outcome is that the oil will take in the flavor of the garlic and onion.
So, by analogy, Paul Ryan has not taken in the flavor of foreign policy, i.e. he is inexperienced in foreign policy.
Readers will probably understand this analogy if they understand the more commonly used analogy of marinating.