I posted a poll on the side to see what topics people like best. I have my hunch, but let's see what you have to say.
Getting back to our story, why is it that you don't say aseh doche lo ta'aseh to allow stolen matzah?
The rule of aseh doche lo ta'aseh only applies when the kiyum mitzvah and violation of the lav are simultaneous (though see Piskei Tos. at the end of Zevachim), but I headed off that problem in the question by asking you to imagine a scenario where the theft and kiyum occur at the same time, e.g. you eat the matzah right off the table without picking it up first.
Another technical possibility mentioned in the comments is that matzah may require ownership, lachem (whether this is true or not is worth discussing, but grant the assumption). Since the aquisition of the matzah, the ownership, did not precede the kiyum hamitzvah, this scenario does not work.
The most popular answer was along the lines suggested by R' Shimon (Ch. to Mes. Nedarim). Aseh doche lo ta'aseh works when the aseh and the lav are isolated to the same person. For example, the Torah does not want you to wear kilayim, but the Torah's desire for you to fulfill the miztvah of tzitzis overrides the negative. When it comes to gezel, it's not just that the Torah wants you to avoid stealing, an issur gavra, but the Torah's aim is to protect the property of your fellow man, the nigzal. Your mitzvas hagavra can override your issur gavra, but not the harm caused to the nigzal. (Notice how R' Shimon's answer assumes something about the reason behind the issur, the sibah. Brisk would not be happy.)
Another example of the same principle at work: the gemara tells us that a neder can be chal on sukkah and prevent your fulfillment of the mitzvah. The Rashba asks: let the mitzvas aseh of sukkah be doche the lav of violating the neder? R' Shimon explains that neder is an issur cheftza, meaning the purpose of neder is to place the object off limits. The mitzvas hagavra of sukkah can override the personal chovas hagavra issur of breaking one's word, but it cannot make an object which is off limits suddenly usable.
No one got the idea suggested by R' Amiel, which is really built on a yesod of R' Shimon in Sha'rei Yosher. Why do we say safeik mamon l'kula - i.e. hamotzi m'chaveiro alav hara'aya, if you want my money, prove it's yours, but until then I am not turning it over. Shouldn't I be concerned that I may be holding money that is really yours, and sfeika d'oraysa (of gezel) l'chumra so I should turn it over?
The key to the answer is knowing which is the cart and which is the horse so we know what comes first. Am I the owner of an object because there is an issur of gezel on anyone else taking it, or is the issur of gezel a function of my ownership? R' Shimon opts for the latter definition. First clarify the rules of the marketplace, what ownership means, how property is aquired, etc. and then we can apply the rules of issur v'heter. If the rules of commerce establish my ownership of an object, I need not be concered with a safeik issur gezel because of someone else's claim. R' Amiel calls this the realm of the mishpati, commercial law, which is distinct from all other areas of halacha.
Aseh doche lo ta'aseh works in the realm of issur v'heter, but not in the realm of the mishpati. There is no commercial meaning to the mitzvah of matzah which would allow me to trespass on another person's property. This idea has a lot of ramifications, but enough for one post.