Graduate math student migrating to engineering? I don't know how to start, truly, but lately I've been involved with some projects in pure mathematics and I've grown frustrated and disappointed. I don't think that pure mathematics is so inspiring; my intent is to move on to study some applied math but most teachers in my department aren't that dedicated to it. All I see are theoretical approaches to problems "that applied sciences may be interested some day".

By talking to a colleague of mine who is seriously considering moving on from Math to Physics, he told me of some engineering departments who might accept someone with a master's degree in their Phd programs. There are some areas I like that might have a connection with what is studied in some Engineering Phd programs, like applied ODE and Dynamical systems and I'd love to put some programming on what I do.

But every teacher I talked about this so far tells me to avoid this kind of migration, since it brings a loss of academic focus and it might be hard for me to find a job later as a teacher. There is no strong tradition in Mathematical Engineering as there is in other countries (I know that such courses exists in Italy and some other European contries).

So I'd like to know some thoughts from more seasoned students and researchers out there. Any advice is welcome.
The only problem with switching to a particular field of engineering is that you'll be missing a core set of knowledge particular to that discipline that will hinder your progress for a while.  For instance, you would probably do great in many electrical engineering sub-diciplines, but if you've never taken a circuits or signals course, you're going to be lost for a while before you can learn that material.

If you already have research experience, that will count for a lot, but I would suggest finding an engineering program that would allow you to take a few undergraduate courses during your first year or two in order to get up to speed.  If you find a professor that is looking for a mathematics-savvy graduate student to fill in the gaps in his/her lab, that might be the best way to get your foot in the door.  You've got skills that could be very valuable to the right lab, and if you use those as leverage to join a particular lab/research team, you'll be able to fill in the missing pieces to your new field and move forward with your own degree.