Shalom! This is an interesting if difficult question. For one thing, are the Lord's Supper and Pesach/Passover the same thing? My understanding of the Lord's Supper is that it was something done after the Pesach meal - it was not a modified Pesach. Please see my sermon at:

http://nccg.org/mlt/sermons/3_149.html

which goes into some detail on this important subject.

I used to think along your lines until I realised that the Passover 'Freedom Feast' was not the same as the Lord's Supper even though they are obviously connected and related. There are two types of bread involved - one depicting human life which of necessity must have the yeast of sin removed from it, and the Divine Life of Yah'shua in which there was no sin. Yeast/leaven is used scripturally in two ways - (1) negatively, of sin; and (2) positively of the Kingdom of Elohim (which is likened to a little leaven). Yah'shua is the Bread of Life, fully leavened with the Kingdom, because He is the Kingdom.

Our belief (unless contrary evidence can be provided) is that passover bread (matzah) should be unleavened but Eucharist bread should be leavened.

I understand that Passover and the Lord's Supper isn't the same... I'm not sure though (if the Lord's Supper was instituted after the Passover meal) where they would have gotten leavened bread from as people started to clean their houses of all the leavened products for the Festival of Unleavened Bread that followed Passover?

A valid point - and on this occasion (on passover itself) I am sure you are right in asserting that only matzah was available for this Last Supper. By the time we get to 1 Corinthians and Paul's commentary on the Eucharist we are seeing a development that includes serious self-examination so as not to pollute the one partaking. Moreover, as my sermon points out, this probably followed an ordinary Agapé meal which, if not on Pesach or during Chag haMatzah, almost certainly meant the use of leavened bread.

Of interest would be a study of the use of different types of bread in daily Israelite life. Which (outside of the Pesach week) was more typical - leavened or unleavened bread? Did your average household use both or was one more typical? Which was preferred? And why? What do we know from the earliest Chrsistian writings (before the Church became hostile to Torah in the nascent Catholic days) on the type of bread used in the Eucharist? This would be interesting to find out and might shed light on the question you raise.