It depends?

Henry II of England was loosey-goosey, but he was a great administrator and had to rebuild a lot of the country's adminsitration after the Matilda / Stephen civil war. There were ministeriales, office holders, but his habit was to grab people and stick them in jobs. Warren's Henry II gives a theoretical org chart, but then writes that it's of limited use. It might look something like that from the outside, but it wasn't how it worked in practice, especially at the top. P. 308 in U. Cal. edition:

If one looks to the men instead of the offices, it becomes difficult to classify them, such is the variety of work they perform and titles they bear. How is one to classify Ralph FitzStephen? He appears as a witness to many royal charters, dignified with the title of "chamberlain": he was only one of the chamberlains, for there are several who bear the title of camerarius simultaneously. He is to be found ferrying money across the Channel, taking charge of the baggage of the duck and duchess of Saxony when they visited England, attending to the needs of the captive Queen Eleanor, or supervising the building of the house of Carthusian monkes which the king founded at Witham - all of which duties might, conceivably, fall to himas a camerarius. But he is also to be found as sheriff of Gloustershire for several years, very frequently as an itinerant justice [Henry II pressed a lot of those around him into that service], as custodian of part of the honor of Richmond, and as the assessor of taxes on the forest of Sherwood and Chippenham. If one speaks of Manasser Bisset, is it as steward of the household or baron of the exchequer and itinerant justice? Or of Richard du Hommet: sheriff of Rutland for most of the reign, but constable "of Normandy" too, from 1154 to 1180.

There were also men without office title who frequently hung about the king - part of his familiares. An example was Hugh de Cressy: took charge of Rouen in 1180, itinerant justice, "but he appears little on the rolls and might be tought of small ccount, were it not that he appears time and time again in the witness lists of royal charters."

Other men were restricted. Richard FitzNigel was treasurer, and doesn't appear doing anything else. William Basset was only a baron of the English exchequer, except for witnessing royal charters when the king was in England. But the higher reaches tended not to have specialists.

Consider the etymologies of what later became great offices. The chancellor was originally the guy who ran the royal chapel. Constable, who could be in charge of the army: comes stabuli, "count (head) of the stable". Butler: became some sort of royal officer in Scotland, but was originally in charge of the bottles in the pantry. Steward: originally "stead warder", became effective head of state in the Netherlands and great officers in Britain.

My impression in medieval Europe -- and I'm certainly willing to be corrected! -- is that royal administration in the earliest days was based on men who were the king's familia, familiares, people about the king, doing local duties for the king, whose duties were changes and expanded or abolished into greater officers. I believe bureaucratizing, narrow jobs, "ministers" in your sense, happened as centuries went on, and royal governments expanded in reach.

Just a clarifying question for our panel: Do you mean “ministers” like cabinet ministers in parliamentary government, or “ministers” like priests/pastors?