Where to have PCB with microcontoller designed?
You want to outsource design and manufacturing.  Then you are looking for an engineering firm that provides solutions or consulting.
They will do the entire design and validation for you, often also production.  The ownership of the intellectual property is negotiable, this is a big factor in the price.

There are many of these companies around, they are often small. But some are big.
Some single market companies can also do this as a side project, they'll have the skilled people already.

You can also choose to hire someone temporarily via an engineering consultancy firm. (I don't know the English term for this) 

Network on your linked-in, a few people from your uni years probably landed at one of those companies.