Europe is a very large area.

In the early middle ages Europe contained many provinces ruled by the eastern section of the genuine original Roman Empire, what is usually called the "Byzantine" Empire.

So the "Byzantine" Empire certainly had units of totally professional soldiers.

But it may also have had semi professional soldiers in the theme system. I have the impression that the theme soldiers held farmland through military service and had to serve in the theme army when called to active duty. Thus they were part time soldiers like militia, national guard members or army reservists in the contemporary USA. I am not entirely certain that is the contemporary understanding of the theme system but if it isn't it can be corrected.

And I don't think that all medieval knights all over Europe all during the millennium that the Middle Ages lasted for were always professional soldiers.

Knights and men at arms who resided in the castle of their lord as part of his retinue were professional soldiers who worked for their lord for room and board and maybe cash pay.

But it seems to me that a knight who had a manor and was the lord of the manor would usually hold the manor as a fief from a higher lord. Thus he would administer his manor full time but would have to serve in his lord's army a specified number of days per year and fulfill other duties specified in his feudal agreement. Thus knights who had manors were part time warriors like militia, national guard, or army reservists in the USA.

And I suspect that a number of medieval societies had social classes vaguely similar to knights and thus being semi professional part time warriors.

Most nobles had some permanent staff. It wasn't just a baron keeping peace in his castle and the whole demesne all by himself.

Retinue

Permanent staff in a noble estate was called Retinue. They would include men at arms who's main job was to always be ready to fight for his master.

Town guard

Most medieval cities and bigger towns had their own militias or guards. In peacetime they acted as police or custom officers. In times of war some city troops may be called by their overlord to join the common cause.

Pieter Geerkens is making a good point in the comments: a knight was usually supposed to lead a small unit from his manor/castle/etc. Besides the knights or semi-professional soldiers there could be carriers, helpers, servants, etc. moving with the army. An inportant point about the answer: As the call to arms was often periodic or even annual, it would make more sense if usually his subordinates were the same people and not random totally untrained peasants.

I will illustrate this with XIII c music (unfortunately not early middle ages) This page has commented lirics of the music 'O que da guerra levou cavaleiros' composed by the king Alfonso X. His basic idea was to ridicule the knights who did not fulfill their feudal contract in wartime. Each verse ridicules a specific case (and some historians believe that each verse really correspond to an specific knight). So, let's see how the king was displeased with some of his knights:

O que tragia o pendom sem oito       
The one who brought his banner without eight (without his 8 promised companions)   

e a sa gente nom dava pam coito
and to his people did not give baked bread

nom vem al maio
do not come to the May


O que tragia o pendom sem sete
The one who brought his banner without seven (without his 7 promised companions)   

e cinta ancha e mui gram topete,
and a large belt and well combed hair

nom vem al maio.
do not come to the May

Up to now we have 2 verses about knights who showed up with no companions. The King made them the favor of specifying in the music exactly the number of expected soldiers - besides telling everybody about their greed and vanity.

O que tragia o pendom sem tenda,
The one who brought his banner without tent
(without his military tent and equipment for long term stay. Was he supposing the war would be quick?)

per quant'agora sei de sa fazenda,
considering what I know now about his situation
(probably the King got word that he was rich enough to pay for his equipment)

nom vem al maio.
do not come to the May.

Now the king ridicules another greedy or coward knight who did not bring equipment to stay for the whole annual campaign.

About the 'do not come to the May': It means the feasts of May, a merry party time also including religious Marian feasts. It also means the period when the army must assemble to the main annual campaigns. So telling them not to come to the May means that they do not deserve neither to party and pray with the King nor to fight with the army.

A good moral of this history is 'do not try to fool a King-poet': he will shame you in music for the ages...

PS: If you wonder about the language it is Galician-Portuguese - it is surely not modern Portuguese, so forgive me any mistakes in translation. Comment if you know better.