The Kaf HaChaim (Orach Chaim 583:4) and Ketzeh HaMeteh (s.k. 4) both write: Those who have the minhag to dip their hamotzi into honey on Rosh Hashanah, should make sure not to be mevatel [forgo] on the minhag of the Arizal to dip the hamotzi into salt three times, למתק את הגבורות כידוע. It is not fitting to miss out on this by any meal, and certainly not during these days.
In Shu”t Salmas Chaim (siman 232) they bring from R’ Yosef Chaim Sonnenfeld that one shouldn’t dip his hamotzi into honey and salt at the same time, as honey with salt will be נותן טעם לפגם – have a bad taste, and it’s not fitting to do this on the night of Rosh Hashanah, as the main reason we dip into honey is a good sign, as honey is sweet. Rather, one should first dip one side of the hamotzi into honey and taste it, and then afterwards dip the other side into salt and taste it.
The Siddur Yeshuas Yisrael in the Pirush of the Divrei Chaim says the opposite, and says, first dip into salt three times, then eat a bit, and then dip the rest of the piece of hamotzi into honey.
Similar to this, it is brought down in the Machzor Divrei Yoel that the Satmar Rebbe was accustomed to dip one side of the piece of hamotzi into salt, and the second side into honey, and then he would cut the two pieces in half and eat them together. See further the sefer Minhag Yisrael Torah (583, pg. 83).
