Because the sphere is not homeomorphic to the plane, something's gotta give. The trick is to place the bad point (or points) of the projection at places of little interest. If you're focusing on Europe and Asia and routes between, you want to cover all of the northern hemisphere, including around the pole, and maybe some parts of the southern hemisphere. If you don't want to break your projection--and breaking will visually interrupt representations of routes--then consider an azimuthal equidistant projection centered at some point within or near North America. If you want a conformal projection, so that angles are locally preserved, you should look at some aspect of the stereographic projection. (The usual polar stereographic projection might do the job nicely, without modification.) Both of these can project the entire world (minus one point), but of course both of them introduce fairly grievous distortions at locations near the missing point. They both do fine for a hemisphere and a little bit more.
If you must have Asia to the left and Europe to the right, you are almost forced to break the projection in mid-Asia. Many of the conic and polyconic projections will do a good job of this. For an example that places North America in the middle, see this image.