Boeing and Airbus make most airline category aircraft.Then you have Dehaviland,cessna,bell,etc etc.
Then there's Mcdonnel Douglas who make F14's,along with other other military aircraft.
Only scratching the surface of possible employers.
Why not write and enquire about cadetships and the like.Plenty world wide to choose from.University degree would be minimum qualifications to gain. Also companies like hamilton make MEC's(main engine controller-a sophisticated Carbureter),propellors,hydraulic pumps an the list goes on......
The basic thing has already been designed,it's now a matter of efficiency.
Good luck.