Degrees from Europe are very transferable. Germany, Italy, France, the nordics etc. all offer engineering degrees in english basically for free (or close) for all EU citizens. I cannot think of any better and cheaper education path. Some of the best engineers (and professors!) in the US actually got their education from minor and major Universities in Europe where they got their education for free.
I think you now have to pay for education if you come from outside the EU, at least in France. It is still cheaper than the US because it is massively subsidized by the working class. A person coming to EU only for free /cheap education is not welcomed.
It could change but, so far, most universities haven't applied that rule (even though they could). Even then, we'd be talking 2800 to 3800€ per year, a far cry from $20k.
I wouldn't recommend France, though, because engineering is mainly not taught in universities but in so-called Grandes Ecoles (they're engineering schools), which span years 3-5 and are only accessed after competitive exams at the end of year 2, from a sort of boarding high-school++ program (Classes Préparatoires).
For theoretical studies or even applied fundamental disciplines (like applied math), it's a different story, universities are excellent and they're easy to access (I mean, the better ones are selective, but you could get on board at any time using other degrees, unlike Grandes Ecoles) and get an internationally recognized degree from.
EDIT: Hadn't noticed OP is from the EU, in that case they would pay the same as French people, that's around 500€ per year max.
This is interesting, I only ever thought about my home country but since it's a part of the EU I could attend another EU country's college for cheap?
Yeah, you could go to e.g. TU Delft for 2500€ per year, tuition is often priced by EU/non-EU. That's basically CalTech or MIT level education in engineering for 1/30th of the price. You'd have to get in, of course.