All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
The hadeeth which you referred to, that says that Jibreel (Gabriel), may Allah exalt his mention, used to revise the Quran with the Prophet, sallallahu ‘alayhi wa sallam, every year and that he revised it with him twice on the year of his death is an authentic hadeeth that is cited in Saheeh Al-Bukhaari, Saheeh Muslim, and other Hadeeth books. However, this hadeeth does not indicate that it is obligatory to recite the whole Quran once or twice a year as you mentioned.
You should know that the right of the Quran on the Muslim is to recite it, revere it, act upon its teachings, and refrain from abandoning it in any form of abandonment. It is better, for the one who has the ability, to recite the whole Quran in a month or less, down to a minimum of three days. The Prophet, sallallahu ‘alayhi wa sallam, said to ‘Abdullah ibn ‘Amr ibn Al-‘Aas, may Allah be pleased with him, "Recite the whole Quran once a month." ‘Abdullah said, "I can do more," and he continued to say that until the Prophet, sallallahu ‘alayhi wa sallam, said, "Recite it in three days." [Al-Bukhaari and others]
The maximum period during which the Muslim should recite the whole Quran is twice a year – at least – according to some scholars. The Shaafi‘i scholar Al-Hindi wrote, "The Muslim should recite the Quran twice a year if he cannot recite it more than that." [Fat-h Al-Mu'een]
Abu Haneefah said:
"Whoever recites the Quran twice a year has fulfilled its due right over him."
Imaam Ahmad said, "It is disliked to recite the whole Quran over a period of more than forty days without a valid excuse, because ‘Abdullah ibn ‘Amr ibn Al-‘Aas asked the Prophet, sallallahu ‘alayhi wa sallam, in how many days one should complete the recitation of the Quran, and he answered, 'In forty days.' Then he said, 'In one month.' Then he said, 'In twenty (days).' Then he said, 'In fifteen (days).' Then he said, 'In ten (days).' Then he said, 'In seven (days)...'" [Abu Daawood, At-Tirmithi, and An-Nasaa'i]
Thus, you come to know that there is no text – as far as we know – which specifies that the Muslim should recite the Quran in one year, and that whoever does not recite it in one year does not bear sin for that, but he misses the great reward for it. Likewise, the one who recites a specific chapter every day, there is no harm on him, but he misses the reward of Khatm (reciting the whole Quran). Also, it is not an obligation to recite the whole Quran during the month of Ramadhaan or any other month. Rather, it is recommended to recite it frequently, and it is better to recite the whole Quran once or more in Ramadhaan.
As for your question: "Are there any authentic ahaadeeth which state that the Prophet, sallallahu ‘alayhi wa sallam, commanded us to recite the whole Quran once a year?" We could not find any hadeeth with such a command. However, it was authentically reported that the Prophet, sallallahu ‘alayhi wa sallam, said that reciting the whole Quran should be done in forty days and less, down to seven days, as in the above-mentioned hadeeth of ‘Abdullah ibn ‘Amr, may Allah be pleased with them.
In brief, the answer to these overlapping questions is that the Muslim is not obliged to recite the whole Quran in a specific period of time; rather, it is recommended to recite it frequently according to one's ability. It is well known that people's abilities differ in this regard. Imaam An-Nawawi wrote, commenting on how often the Quran should be recited in full:
"The best view is that it varies from one person to another. The one who is seeking to understand it and ponder its meaning should limit himself to as much as he can understand fully when he reads, and the one who is busy with what is more important, such as spreading knowledge or other than that, should limit himself to what will not cause him to neglect his work. If he is not among the categories mentioned here, then he should do as much as he can without reaching the point of boredom or hathramah (i.e. reading the entire Quran too hastily)." [At-Tibyaan]
Allah knows best.