There were no Universities in the Roman Empire. At that time advanced teaching was very much a one-man show. Teachers would set up their own little schools and collected a small group of students to instruct. They might also have written a book as a record of their teachings for future reference, or perhaps one of their students might produce a written account of the master's words, but in those days, long before printing and word processors, writing a book was a major enterprise. Not many were written and even fewer have survived.
After the collapse of the western half of the Roman Empire teaching of any sort became rare (in western Europe). The main driving force for learning was initially religion, both Christian and Islamic. The churches and mosques required scholars to research and record the faith. These scholars searched out the old works and naturally became interested in the mathematics and philosophy of the ancient world. As things settled down trade began to be restored and attention turned to other major problems, in particular to medicine.
By the 9th century groups of students were once again collecting under chosen teachers. These students were often not young, but older people who needed to learn arithmetic to help in trading, medicine to help deal with sickness in their communities, or priests being trained for higher posts in religious organisations. Learning was often student-driven. Groups of students would organise themselves in schools and look for experts they could hire to teach them. We might define a University as a large school, with more than one teacher, covering several different subjects, which had students from all over Europe.
Within this definition there seem to be three candidates for the claim of being the first university in Western Europe. There was a medical school, which grew up (I am reluctant to say it was 'founded') in the 9th century in Salerno in southern Italy. This specialised in medical subjects and faded away within a few hundred years. Another similar school appeared in Bologna in the 11th century, but this one expanded to cover other topics and exists to this day as the University of Bologna. Finally there was a famous school in Cordova in southern Spain. Spain had been invaded by the Moors in the 8th Century and they rapidly conquered most of the country up to the Pyrenees. By the middle of the 9th century their capital city, at Cordova was probably the largest and most magnificent city in Western Europe - it even had street lighting! It had a large school, which had the great advantage of being in touch with the scholars in the Middle East and, as a result, had access to much of the work of the ancient Greeks. As a result, in addition to religious topics, mathematics, philosophy and medicine were all studied in Cordova. Islam was then a more tolerant religion than Christianity, and the presence of 'unbelievers' was at least tolerated (Christians and Jews were regarded as people who worshipped the same God but had got some misguided ideas), so it was possible for students from all over Europe to study at Cordova and it became a great international school. Personally, I would rate this as the first true University in Europe.
However, civil wars weakened the Moorish rule. The Christians started to regain Spain, and by the 11th century they had re-occupied most of the country. The libraries at Cordova and Toledo were seized and their books found their way into other parts of Europe, stimulating study in the developing Universities there. About 50 years after the University of Bologna had appeared a similar institution was founded in Paris, also run by its students. The University of Paris became the most famous and most influential University in Western Europe and many of the other old European Universities, including Oxford and Cambridge, were modelled on it.
However, the teachers were still not called 'Professors'.