Is there any particular reason the procedure for washing hands for bread includes pouring the water twice on each hand? Isn't once enough?
The basic idea is that the first splash of water becomes tamei from touching your hands (since they themselves are deemed tamei - that's the reason we have to wash in the first place); the second splash, then, makes the water left on your hands tahor. If your hands were dirty, you actually need three splashes: one to remove the dirt, and the second and third serve as the necessary two splashes.
Actually, the Shulchan Aruch in OC 162:2 (agreed upon by the Rama) rules that you need to splash once per hand plus:
one more splash if you are pouring less that a revi'it (86.4 mL) per splash in which case you need to splash twice per hand: once for the washing and once to remove the tamei water. (If you used a full revi'it than the water never becomes tamei from the hands.)
one more non-primary splash at the beginning if you have dirt on your hands, in which case you splash an extra time first to wash off the dirt.
I find that when using less than 86.4 mL per hand it is exceedingly difficult to cover the entire hand with the water. Additionally, a standard washing cup holds about a liter (about 12 revi'its) so the vast majority of people do not need a second splash the way we are accustomed to washing.
I note that the Mishna Berura there (sk 21) notes that it is possible that the Gra holds like the Rashba who requires two splashes always, but the Mishna Berurah agrees that the majority opinion is as I quoted above.
Instead of jumping to the Shulchan Aruch to understand the Why's of Halacha, I would suggest checking out the Gemara. It seems clear that the Gemara is asking why we wash at all. Clearly the minhag to wash was well known but the reason was not.
One answer is “Serach teruma” - that it was made a general custom so that Kohanim would not eat teruma with hands that may have touched impure things. Here too it is not clear if impure means unclean or some form of tuma and other commentators come to answer this.
Another answer is "Mitzva" which is interpreted as following from Kdoshim Tihiyu - a general command to act with sanctity - cleanliness. Considering that the Rabbis compare eating without washing to relations with a prostitute - it seems here also that the focus is on physical cleanliness.
Thus it is generally accepted that the minhag to wash hands derives from a need for physical or spiritual cleanliness before eating bread. The question of why the minhag is not applied to other foods and even wet fruits and vegetables where there is a source for such a custom is also not clear.
Given that it is not clear why we wash hands at all, I would say that it is also likely not completely clear why we wash the way that we do.
The Shulchan Aruch chooses to emphasize the tuma aspects of washing hands and expands the Halachic ramifications of this. It could be that the Shulchan Aruch is deciding the definitive "real" reason for washing or it could be that he is trying to cover all the bases. Indeed as has been mentioned here, if ones hands are dirty, dealing with ritual impurity is insufficient. Therefore it seems to me that the SA is not deciding the dispute in the Gemara and mentions tumah as only one of the things that the washing should cover. In other words, the Why is not Halacha. The How is Halacha and the Why is still an open question.
While not disagreeing with any of the other answers, I would like to suggest another possible answer as to the original Minhag. If you pour only once onto your hand with a cup, it is likely you are pouring on only one side of your hand. Note that if this is indeed the reason for pouring twice, it would mean that we should also turn our hands over before the second pouring.
The Biur Halacha expounding on the Tumah view, similarly notes that one could consider a third splash necessary in order to ensure that the whole hand is reached (to remove the water of the first splash).
In short - I think the original reason for 2 splashes is open to various answers depending on if the original minhag was based on cleanliness, tuma, or both and this is completely compatible with Halacha.