Read
this thread and the advice given by the member rudestuff. It's about training both in the US and EU. You can complete the training with a cost up to 40-41k. And since you're a US citizen, you won't have to search for schools that are approved to train foreign students (SEVP) - which tend to be more expensive. Poland, Czech republic, Hungary are also an option if you're looking for cheap training.
I think holding permanent residency for Finland does not give you the right to work in other EU countries but only in the issuing country. I assume that you can work on acquiring the Finnish citizenship which will then open doors in the rest of the European countries. Modular and distance learning is the way to go if you're on a budget.
But IMO, given that you have the right to work in the US, If I were you I would consider going back there. Better job market, salaries etc.