It is not at all clear one needs to use a utensil to wash hands after using a bathroom. In a review of sources, R Ari Zivotofsky writes (here)
Misconception: Upon exiting a bathroom, one should wash by pouring water from a keli (utensil) three times on each hand.
Fact: There is an obligation to wash after using the bathroom (Shulchan Aruch OC 4:18), but three times is not specified, and a keli
is not required.
Details: The earliest source for washing upon exiting the bathroom is a mishnah
(Yoma 28a) that states that in the Beit Hamikdash there was a rule
requiring a Kohen to immerse in the mikvah after he defecated and to
sanctify (i.e. wash) his hands and feet with water from the Temple’s
laver after he urinated. The mishnah seems to portray this as a
rule unique to the Temple. The Talmud (Yoma 29b-30a) explains that the
requirement for washing hands and feet is solely for hygienic
purposes. Rabbeinu Tam (ibid, Tosafot Yeshanim, s.v. mitzvah) deduces
that if a Kohen urinated but his hands remain clean, there is no
obligation to wash.
All the sources that mention a non-Beit Hamikdash-related washing
after the bathroom are post-Talmudic. These sources include the
Tashbetz, Kolbo and Mordechai, all of whom are cited by the Beit Yosef
(OC 4). Other than cleanliness, there are two reasons discussed for
the washing. Similar to washing before davening (OC 92:4), some view
it as a preparation for reciting the berachah Asher Yatzar after using
the bathroom (see Tur OC 165 and commentaries). Additionally,
according to the Mishnah Berurah (4:40; 227:11) and others, merely
entering a beit hakisei (an old-fashioned as opposed to a modern
bathroom) imposes a ruach ra, which must be removed by washing.
However, the mishnah regarding the Temple referred to above presumably
refers to a Kohen who took care of his needs in a bathroom and yet
the Talmud makes no mention of a ruach ra. Furthermore, many
authorities argue that a modern, clean, multi-purpose bathroom does
not convey a ruach ra even if an old-fashioned bathroom does.
The Aruch Hashulchan (OC 7:3) summarizes these sources as follows: If,
while using the facilities, one did not touch anything dirty or any
part of his body that is normally covered, there is no requirement to
wash at all. Nonetheless, the custom is to wash, either because of
cleanliness or because it is proper in order to say Asher Yatzar. He
observes that some have the custom to wash thrice, but the Beit Yosef
does not rule that way. The Aruch Hashulchan makes no mention of a
keli or ruach ra.
According to those who hold that the washing is for cleanliness or in
preparation for Asher Yatzar, there are few ritual requirements.
Similar to the washing prior to davening, such a washing requires
neither a keli nor three washings. According to the Mishnah Berurah
(4:38), which holds that the washing is due to a ruach ra, the washing
still has fewer rules than other ritual washings — such as for bread — and
does not require a keli (Mishnah Berurah 165:2). The washing after
using the bathroom is more akin to a washing for hygienic reasons,
such as is required after removing one’s shoes or cutting one’s hair
(SA OC 4:18).
The Yalkut Yosef similarly indicates that washing
after using the bathroom does not require a utensil. This was
stated explicitly by Harav Ben Tzion Abba Shaul who reportedly washed from a faucet even when a utensil was
readily available, in order to emphasize this point.
Rav Yitzchak Yaakov Weiss observes that while
most authorities state that a keli is not required, in deference to
the minority opinion, it is a good idea to use a keli, although he
reiterates that it is not required.
In the yeshivah world washing with a utensil after using the bathroom
is a widespread practice, possibly based on the custom of the Chazon
Ish as publicized in Ta’ama D’kra. However, this custom seems to
have been a personal stringency of the Chazon Ish and not intended
as a halachah for the masses.
See the original for more details and sources.