The PUFA's in birds go rancid very quickly, so pressure cooking may be a better option. Furthermore, rosemary and oregano are known to inhibit oxidation of PUFA's during cooking.
Here is a patent describing the effects of various herbs on PUFA oxidation alongside their extracts:
http://www.pat2pdf.org/pat2pdf/foo.pl?number=4525306
https://www.ncbi.nlm.nih.gov/pubmed/18034711 said:Abstract
Capabilities of methanol extracts from oregano and rosemary in retarding oxidation of long-chain polyunsaturated fatty acids, docosahexaenoic acid C22:6 (DHA) and eicosapentaenoic acid C20:5 (EPA), in menhaden oil were investigated. The fish oils after mixing with the extracts at different concentrations were oxidized in an accelerated study by heating at 150 degrees C for 30 min or incubating at 60 degrees C for 5 d. After heating at 150 degrees C, only 15.9% of DHA and 18.5% of EPA remained in the fish oil without extract, while 38.8% to 65.9% of DHA and 44.7% to 69.0% of EPA were retained in the fish oil mixed with 1% to 5% of oregano extract. The highest retained DHA (56.9%) and EPA (58.0%) in the fish oils mixed with rosemary extract were observed at 2.5% addition. Increasing rosemary extract to 5% lowered its capability of inhibiting DHA and EPA oxidation. After incubation at 60 degrees C for 5 d, the highest inhibition capability was also found at 2.5% of added rosemary extract, and the oil retained 88.2% DHA and 88.3% EPA. However, only 18.8% DHA and 23.6% EPA were retained in the fish oil mixed with 5% of oregano extract and no DHA and EPA were detected in the fish oil without extract after 5-d incubation at 60 degrees C. Thus, antioxidant activity of the rosemary extract was greater than that of oregano extract, but was sensitive to heat. The rosemary extract also demonstrated higher DPPH (2,2'-diphenyl-1-picrylhydrazyl) free radical scavenging capability, which was approximately 3 times higher than oregano extract, although there was no significant difference in the total phenolic contents between both extracts.
Here is a patent describing the effects of various herbs on PUFA oxidation alongside their extracts:
http://www.pat2pdf.org/pat2pdf/foo.pl?number=4525306