Relative to interrupting between Netilat Yadaim and HaMotzi, the laws of interruption for after HaMotzi are stricter. Just like any Birkat HaNehenin(blessings that are recited over foods, fragrances, etc.), one must recite HaMotziand then immediately eat the bread. The Gemara (Berachot 40a) discusses two exceptions to this; if one needs to ask for something that is connected to the meal, such as bringing salt to the table, then one may do so after reciting HaMotzi. Additionally, there is a Halacha that one must feed one’s animals before eating, and therefore if one needed to feed ones animals it would be permitted after reciting HaMotzi.
Another example of a permitted interruption is when one recites “Sheheheyanu” on a new food. One first recites the appropriate Birkat HaNehenin, thenShehehyenu and then partakes of the food. Since Sheheheyanu is connected to the first blessing it is not considered an interruption. The same applies to recitingHaMotzi and then the blessing of “Leshev BaSukkah” and then eating bread. As well, the Tur discusses a custom in the name of Rav Amram Gaon in which one recites the blessing of HaGefen, then a series of oaths, and ends off with the blessing of Hamavdil. These oaths were connected to the theme of Havdala and therefore were not considered an interruption.
There is a Moroccan custom for the head of the household to recite HaMotzi, then dip the bread in salt while reciting “Hashem Melech, Hashem Malach, Hashem Yimloch LeOlam Va’ed”, before eating the bread. Rabbi Shalom Messas (Shemesh Umagen), Rabbi David Ovadia (Nahagu Ha’am) and Rabbi Yeshoshua Maman all wrote that these verses are considered an interruption and should therefore be avoided. Nevertheless, Rabbi Moshe Malka (Shu”t Mikve HaMayim) writes that these verses are not an interruption and may be recited. Furthermore, Rabbi Michael Peretz defends “Hashem Melech” as an established custom connected to HaMotzi just like the oaths of Rav Amram Gaon are connected toHavdala as well as the aforementioned examples, and not a haphazard recital of verses.
Summary: One should not make interruptions between the blessings ofHaMotzi and eating bread. There are opinions permitting the custom of reciting “Hashem Melech”.