On rare occasions, the fast of Asara Be’Tebet falls on Friday. Normally, there is a special Misva to taste the foods one prepares for Shabbat to ensure that they taste flavorful and will be enjoyed on Shabbat. The Rabbis teach that the value of this practice is alluded to in our Shabbat prayer service, where we recite, "To’ameha Hayim Zachu" – "those who taste it [Shabbat] have earned life," indicating that those who taste the Shabbat food when preparing it is rewarded with long life. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes (Parashat Lech-Lecha, Shena Sheniya, 7) that when Asara Be’Tebet falls on Friday, one should observe this practice by tasting the Shabbat foods in one’s mouth and then expectorating them, without swallowing them. As putting food in one’s mouth without swallowing is allowed on fast days, this is the way one can observe this Halacha when Friday is a fast day.
When Asara Be’Tebet falls on Friday, Vidui is not recited after Minha as it is on ordinary fast days. Rather, the congregation proceeds from the repetition of the Amida to "Yehi Hashem" and then to "Hashem Malach," like on a normal Ereb Shabbat.
If one prays at home on Friday night when Asara Be’Tebet falls on Friday, or if a congregation prays early, they must ensure not to eat until Set Ha’kochavim (nightfall). Normally, a Minyan that is held at the usual time will finish the Friday night service after Set Ha’kochavim. But if not, one must remember not to recite Kiddush and eat until after Set Ha’kochavim.
Additionally, every Ereb Shabbat, it is proper to verbally declare before sundown that one accepts "Tosefet Shabbat" – some extra time that he adds onto Shabbat. One must make an effort to remember do this also when Asara Be’Tebet falls on Ereb Shabbat, and thus Minha is longer and might end very close to sunset.
Summary: When Asara Be’Tebet falls on Friday, one should taste the Shabbat foods when preparing them without swallowing. Meaning, one should taste the food in his mouth and then expectorate, such that the food is not swallowed. Vidui is not recited after Minha on this day, as it is on ordinary fast days. One must ensure not to recite Kiddush and begin eating until after sundown.