Life of a medieval university graduate meant, probably, continuing as a scholar in the same place. Doctoral degrees could be in theology, law or medicine. The degree cost a large sum to purchase, so many students never graduated. (They went on to be tutors, magicians, and stand-up comics.)
The only place for someone with a PhD in theology was right there at the University of Paris, so daily life changed little. The most famous Doctor of Theology was Peter Abelard (1079-1142). He rose to fame while the university system was still nascent, but his life as lecturer and debater works as a generic pattern. After defeating competing scholars in rhetoric and dialectic, he studied theology with a famous teacher and became a canon at Notre Dame. His fame as a lecturer came from his debate victories and at his prime, he drew large crowds of students. But as a canon (a very low order of priest, sort of), he was supposed to remain single; Abelard famously fell in love with a beautiful, brilliant teenage girl he was tutoring. Her guardian caught them having sex and all hell broke loose. It’s a complex story, worth telling in less summary, but at the end of it, their child was placed with his sister in the country, she was at a convent, and he was castrated by a gang of hired thugs. He ended his life as the Abbot of a rural monastery, exiled from all he cared about. But in his prime, his daily life was about philosophy, theology, and lecturing to enthralled crowds.
A PhD in medicine fitted someone to be a court physician. Surgeons were trained by apprenticeship, usually on battlefields. Physicians, by contrast, were expected to know the learned works of Hippocrates, Galen and Avicenna. They diagnosed diseases and prescribed spices to rebalance body chemistry that had gotten too hot, cold, wet or dry. Among the known diseases they tried to treat were infectious diseases like St. Anthony’s Fire, smallpox, dysentery, cholera, tuberculosis, and leprosy. Leprosy was a fad diagnosis, badly distinguished from similar skin problems and even syphilis. Court physicians also diagnosed by urine tests. Every doctor had a urine sample bottle; university medical exams required him to know the smell and look of various ailments’ urine. A really well-paid, highly placed royal doctor also stood behind the king’s chair at dinner and advised him what to eat, to maintain the king’s health.
I’m afraid I don’t know much about the life of a trained lawyer. I suspect that a real Doctor of Law was over-trained for real court use, and was left instead to teach philosophy of law at a university, leading a life much like Abelard’s original lecture routine. There just weren’t that many of them; their lives were marginal in the medieval economy. I can imagine that toward the later centuries, Italian merchants and bankers might hire one as consultant for contracts. “Law” meant “Roman law,” part of the fiction that Rome’s Empire never really ended. Designing a contract that met some obscure point of Roman law might ensure victory in a court challenge by a competitor.