The term "Pat Ha’ba’a Be’kisnin" refers to grain products that normally require the Beracha of "Mezonot," but, when eaten in large quantities – 216 grams or more – require the Beracha of "Ha’mosi." One of the definitions given by the Shulchan Aruch (Orah Haim 168:7) for "Pat Ha’ba’a Be’kisnin" is a product baked from dough with some sort of filling, such as nuts, honey or jelly. One who eats such a product recites "Mezonot," unless he eats 216 grams or more, in which case he recites "Ha’mosi."
However, several passages later (168:17), the Shulhan Aruch appears to contradict this ruling. There he discusses a product called "Pashtida," which is made from dough that was stuffed with meat, cheese or fish, and the Shulhan Aruch rules that this product is considered bread, and it thus requires the Beracha of "Ha’mosi." This seems to directly contradict his earlier ruling, that dough with a filling falls under the category of "Pat Ha’ba’a Be’kisnin," and thus requires "Mezonot" unless it is eaten in large quantities.
Two main answers have been offered to resolve this contradiction – which yield very different practical Halachic results. The Taz (Rav David Segal, 1586-1667) suggests, very simply, that the two passages in the Shulhan Aruch refer to different quantities. Indeed, all dough baked with filling falls under the category of "Pat Ha’ba’a Be’kisnin," and thus requires "Mezonot" unless it is eaten in large quantities. When the Shulhan Aruch speaks of the "Pashtida" which requires "Ha’mosi," he refers to a case where one ate 216 grams or more, and for this reason, the Beracha is "Ha’mosi." According to the Taz, then, there is no distinction between different types of baked products with filling, and they all require "Mezonot" unless they are eaten in a quantity of 216 grams or more.
The Magen Abraham (Rav Abraham Gombiner, 1633-1683), however, disagrees, and contends that the Shulhan Aruch distinguished between different kinds of products. When the dough is filled with sweeteners, or with snack foods such as nuts, then the product is not considered bread unless it is eaten in large quantities. The "Pashtida," however, is filled with foods that people eat for satiation – meat, fish or cheese – and, as such, the product has the status of bread in all respects. Therefore, the Beracha over such a product is "Ha’mosi," regardless of how much one eats.
Hacham Ovadia Yosef brings several proofs to the Magen Abraham’s understanding of the Shulhan Aruch’s position. He notes as well that Rav Yitzhak Chehebar (1912-1990), one of the leading Rabbis of Halab (Aleppo), proved that the custom in Halab followed the Magen Abraham’s opinion. There was a certain product made from dough with meat inside which was common among the Jews of Aleppo, and over which they recited "Ha’mosi" – clearly indicating that they followed the view of the Magen Abraham, that if dough is filled with meat or cheese, the Beracha is "Ha’mosi."
A common modern-day application of this Halacha is calzone – dough filled with cheese. As Halacha follows the view of the Magen Abraham, one who eats calzone must wash Netilat Yadayim, and recite "Ha’mosi" and Birkat Ha’mazon, just like when one eats ordinary bread.
Furthermore, Hacham Ovadia asserts that no distinction is drawn in this regard between dough that is filled with meat or cheese, and dough that has meat or cheese on top of it. Therefore, Hacham Ovadia maintained that pizza – which is, essentially, bread covered with a layer of cheese – has the status of ordinary bread, and requires "Ha’mosi." Rabbi Yisrael Bitan (contemporary) writes that when Hacham Ovadia was served pizza, he would wash Netilat Yadayim and recite "Ha’mosi" over regular bread, in consideration of the opinion of the Taz. Nevertheless, as far as normative Halacha is concerned, Hacham Ovadia in his writings clearly accepted the view of the Magen Abraham, and therefore, one who eats pizza or calzone recites "Ha’mosi."
Summary: If dough is baked with a sweet filling, or with nuts, then it requires the Beracha of "Mezonot," unless it is eaten in a quantity of 216 grams or more, in which case it requires "Ha’mosi." However, if dough is baked with something more substantial, such as cheese or meat, then it has the status of ordinary bread, and the Beracha is "Ha’mosi." As such, the Beracha over pizza and calzone is "Ha’mosi."