Q: Is it permissible for Ashkenazim to eat soft Matzah, like that eaten by Sefardim?
A: There are those who forbid it. While they agree that in the past Ashkenazim did eat it, they ceased doing so (Ha-Rav Shlomo Zalman Auerbach, Halichot Shlomo chap. 9 note #80) and there is no tradition (Masoret) to permit it. But there is also no tradition to forbid it, and the Rama (Orach Chaim 460:4) mentions that the Matzot were thick (and must therefore have been soft). And the Sha'arei Teshuvah writes there that they would prepare them with a type of grater, which is also a sign that they were soft. Ha-Rav Asher Weiss writes that the basic Halachah is that they are permissible, but he is concerned that we are not experts in making them soft and there is therefore a concern about Chametz (at the end of his Haggadah, Siman #15). But Sefardim are experts. And Ha-Rav Herschel Schachter also permits them. In sum: It is permissible.