Soft Matzah is Making a Comeback
The following statement can be found in the Shulḥan Arukh, OH 446:4:
מצא פת בפסח בביתו ואינו יודע אם הוא חמץ או מצה מותר אפילו באכילה דאזלינן בתר בתרא.
If during Pesaḥ one finds in his house pat (bread) and doesn’t know if it is ḥametz or matzah, it is permitted even to eat, since we following the most recent [i.e. we are using matzah at the moment-Mishnah Berurah]
From this halakhah it seems as if a person would not be able to tell the difference between pat that was matzah, i.e. kosher for Pesaḥ, and pat that was ḥametz. Not surprisingly, the Mishnah Berurah commented on this and wrote:
דין זה הוא לפי מנהג זמנם שהיו אופין מצות עבה קצת ולא היו חלוקין בתארם מככרות של חמץ
This law is according to the practice of their time when they would bake matzot that were a little thick, and they weren’t distinguishable in their shape from loaves of ḥametz.
It seems as if in some circles soft matzah is making a comeback. For those in America soft matzah can be ordered here. The website writes that:
Since Soft Matza is a little thicker than standard “crunchy” Matzot, there are differing views. In general, people of Sephardic lineage have a minhag (tradition) to eat Soft Matzot on Passover. According to “Maran” (R’ Yosef Karo – Shulhan Aruch) a Matza may be baked up to ‘1 Tefah’ (approximately 3-4 inches) thick. People from Ashkenazic lineage have a minhag (tradition) to eat Matza that is as thin as possible, and therefore should consult their Rabbi to determine if they are allowed to our thicker Matza. You can see from the pictures that each one is about 1/3 – 1/2 of an inch thick.
In Israel Makor Rishon has an article about a bakery in Tzefat that bakes soft matzot.
See this post by Rabbi Josh Waxman for a first-hand experience with soft matzot. Rabbi Waxman’s post addresses the opinion of Rabbi Herschel Schachter regarding Ashkenazim eating soft matzot, and Failed Messiah also brings here a letter from Rabbi Schachter. Rabbi Shlomo Aviner also addressed this question, relying upon Rabbi Schachter’s permissive opinion.