Evangelicals have conflicted feelings about Patristic studies, and that conflicted ambiguity is seen in how the field exists.
What do I mean? In the past few years it has been not uncommon to talk about evangelicals becoming more interested in the patristics. Here's an article that touches on the theme from Christianity Today, Nov 2009. However that "upsurge" is fairly limited, and is almost always "adjunctive" to something else.
No leading evangelical scholar holds a post in Patristics, so far as I can tell. Bryan Litfin's position is in Theology. As is Christopher Hall's.Probably the exception that proves the rule would be D.H. Williams, Professor of Religion in Patristics and Historical Theology at Baylor.
Partly the reason is that Protestants rarely offer a course selection that would ever justify a Patristics staff member. Church History in the 1-500 period is almost always taught as a survey, introductory level course, which means it never gets the depth it deserves (radical thought number one, what if Reformation History was your Introductory course, and Early Church History your 2nd year?). And later engagement with Patristics is more often than not an adjunct to systematic theology (Trinity, Christology if you're lucky). So a Patristics scholar has exactly one course to teach in their specialty, which is why an evangelical patristics scholar will almost never get a job in a seminary, and is likely to end up in the academy at large. It also explains why the most often end up in the Doctrine field, but then Patristics becomes a means to an end: pillaged for supplying ammunition for debates born out of later times. Trinitarian scholars are probably the worst at this. It took me about 5 read-throughs of T.F. Torrance's The Trinitarian Faith to understand it, and then on the 6th time I realised that I thought he had fundamentally misread key aspects of Athanasius and the Cappadocians.
I'm not saying this is a, or even the, problem, I think it's symptomatic. Evangelicals value the Bible, and get trained in the Historical Critical method, and then at their worse think that Patristics is negligible because they were all bad exegetes who got their Doctrine wrong. Furthermore, they are prone to hear the unfortunate apologetic claims of Roman Catholics and Eastern Orthodox churches saying "these are our Fathers" which makes Evangelicals suspicious, and see church Fathers as dangerous/other/possibly a temptation to depart from Evangelicalism (which, admittedly, has happened).
This is partly a double misconception about Patristic Exegesis and Church History. If Protestants only ever define themselves as the offspring of 16th century Roman Catholicism, their claim to Catholicity itself is badly injured. They will forever be reactionary. More than that, one simply cannot accept the claim that Fathers "belong" to the RC or EO communions. Because before 451 there was one church. This is, admittedly, my new slightly idiosyncratic dating of the "Greater Schism", because I think we need to reorient Early Church History and say that from 451 onwards you have 3 major streams: The Chalcedonian Church, the Church of the East (aka 'Nestorian'), and the Oriental Orthodox (aka 'Monophysite'), even though the latter doesn't quite emerge full-fledged at the time, and the former has already emerged, their self-definition in terms of rejection of Chalcedon is the best point to go with. Anyway, my point is that Evangelicals cannot cede 'propriety' of Church Fathers simply because of other churches claims to historic continuity, since such claims are meaningless in an age of undivided Christianity.
Furthermore, there must be greater engagement on the issue of hermeneutics, because any claim to be confessional Christianity built on the great Creeds (Nicaea, Constantinople, Chalcedon) can only be upheld if, to some extent, we mean what they meant. If the Evangelical reading of Church History is "they got doctrine right, but for all the wrong reasons", claims to confessional continuity would be rendered empty at best, dissembling at worst.
Another set of problems is that Evangelicalism is afflicted with a kind of tribalism, and semi-Fundamentalism. I realise that there is actually a very clear distinction between Evangelicalism and Fundamentalism, even if outsiders don't, but sociologically evangelicals have fundamentalism tendencies. Things like guilt by association, guilt by association of association. A troubling insider/outsider dynamic. This impacts Patristics because, as I've said above, it comes with suspicion, and is seen as Other, and the deeper you get into it, the more suspicious and Other you become to the Insider. However "Evangelical" you are, the question will always be, "Why don't you just focus on the Bible?"
I have more to say on this, but I have rambled enough for one day.