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You do not normally get a shortage in the labour market because as price goes up, the appeal of hiring someone goes down. As you would expect, the formal definition of shortage is not "not being able to get what you want", a shortage is defined as "a situation where an external mechanism, like government regulation, prevents price from rising."

There are real cases of shortages in the labour market. Where I live, the law defines a fixed rate for doctors. Even if I had all the money in the world, I could not legally pay more than my neighbour to employ the services of the doctor in town. Here, a shortage is a real possibility as there is no monetary way to decide who gets the appointment with the doctor. But these tend to be exceptional cases.

Without that legal requirement, I could simply offer more than my neighbour to get the appointment and the neighbour would have to choose to go without seeing the doctor. There is not a shortage in that scenario as the neighbour was not in the market to begin with. This is the same reason we don't say there is a shortage of Ferraris, even though the vast majority of people will never be able to get their hands on one, no matter how much they wish they could have one. They are simply unaffordable to most people.

It really doesn't matter how long it takes to train someone. As long as the price is able to rise, the business who really need the existing people will pay more, taking them from the businesses that cannot afford them, until everyone who still needs that service is fulfilled.