The answer depends very much on your understanding of "best constructed". Does this mean the best looking? The best decorated? The tallest? The largest ground plan? The best foundations? The best structural integrity? The most stable? The least rebuilt?
"Best" is always an impossible term to quantify; every observer will have a different opinion about which medieval cathedral is best constructed - and there are hundreds across Europe that could reasonably hold the title.
The same people who built them. Mainly stone masons.
Depends on how big or how much. Typically rock quarries or cliff walls are the common source of stone.
The interior of the medieval cathedral was positively cavernous in its size and stone ornamentation.
There were stone masons and brick masons. They built things out of stone and brick respectively. During the Middle Ages, stone masons were responsible for a lot of stone carving and sculpture, in addition to just building with dressed stone.
Depends. A stone mason generally gathered stone and distributed to those who needed; i.e. constructors, black smiths, etc. However, there were other kinds of masons.
Master masons managed groups of workers building stone or brick buildings. They trained apprentices. They had a role in the design of buildings until the time came when architects dominated this work.
Using natural stone from the earth, stone masons create sculptures, buildings, and structures. Castles and cathedrals gave masons the majority of their employment during the Middle Ages.
They chiselled stone.
Masons
Yes, they had to travel in order to find work. Master Masons were highly skilled (and highly paid) craftsmen who were in great demand all over Europe, but they worked in a dangerous environment: following a major fire at Canterbury Cathedral (Christ Church Cathedral Priory) in 1174, a French Master Mason named William of Sens travelled to Canterbury to begin rebuilding the choir. He later fell from scaffolding and was very seriously injured; he may have been paralysed. The work on the Cathedral was then completed by another mason known only as "William the Englishman", who probably travelled from some other part of the country. Master Masons would not only be responsible for overseeing the building work, they often drew up plans and created the master drawings for the shapes of stones used in columns, arches and other areas - much like a modern chief architect. Under them would be large teams of masons and assistants.
Stone Masons for large stones.
There are hundreds of stone medieval churches in Europe. One of my favorites is St. John the Baptist Church, in Plymtree, Devonshire, England. It is a lovely little country church for common folk of the village. Wikipedia has a nice article listing medieval churches in York, complete with a few pictures, and I think all are stone. There is a link below. Winchester Cathedral is one of many that are well known.