Can DNA sequencing of food provide quantitatively the fraction of protein from chicken vs soy?
While this question falls somewhat between biology, chemistry, and food science, the use of quantitative DNA analysis as a proxy for estimation of relative protein fractions from two different species (in this case plant vs mammal) is probably on-topic here considering it is biologists who have invented, use, and interpret results of DNA amplification and sequencing analysis, and can best speak to their correct use and potential problems when used outside their accepted protocols.
I am not a trained biologist myself, so I can not state with authority that starting with a food product whose plant and animal components may have started from different places on Earth, been processed differently, then combined and further processed involving major excursions in heat (cooking) and in pH deviate substantially from accepted protocols for DNA quantitative analysis, but I have a strong hunch that this is probably so.
As @BryanKrause has noted in a comment, the laboratory in question's primary expertise is in animal identification - does a carcass or piece of meat or fish come from the species claimed or not, and this is a laudable service. But using relative fractions of DNA within a single sample of prepared food as proxy for relative contribution of protein is far outside that kind of service, and as far as I know, is not an accepted practice anywhere.
Ars Technica is a popular technical news source that generally does a excellent job of summarizing scientific topics when they do cover them, in my opinion at least. In an update to the article Subway releases data after scientists weigh in on 50% chicken test [Updated] they summarize released information from two independent laboratories using an &quot;ELISA methodology designed for food products to quantify soy in the chicken.&quot; ELISA is a quantitative method that can be used to measure the presence of various proteins directly, and does not use DNA as a proxy.

Update, March 6: Subway has released the lab reports from both of its independent tests. Both Maxxam Analytics in Ontario, Canada and Elisa Technologies, Inc. in Florida used enzyme-linked immunosorbent assays (ELISA) designed for food products to quantify soy in the chicken. ELISA's are a standard type of assay that generally detect and quantify substances based on binding by an antibody. In the assay, antibody binding kicks off a detectable chemical reaction, commonly resulting in a color change. In the case of Elisa Technologies, the lab used an antibody that binds to soy flour proteins and the lab used known concentrations of those soy proteins for comparison to determine the quantity of soy protein in Subway's chicken samples.
In all samples, Elisa detected 3 parts-per-million or less of soy proteins, which is well below one percent of the chicken. Maxxam detected 5.3 ppm of soy protein in the chicken, which is still well below one percent. (emphasis added)