As an aside, when trying to take out a unit of flyers, or something really big and nasty through the use of a brigade of something. If you can, then make sure that the first unit that goes in does not actually engage the unit being charged, but goes around it to block its retreat. Then pile in with your combat units, etc... to 'seal the deal'.
You can quite easily take out a joint brigade of a dragon and Eagles through the use of a brigade of three Empire knights by aligning the first unit to maximise the 'backstop' area, so that stand removal will be of no use. Then pile the other two units in and concentrate as many of your attacks on the Eagles as you can, and of course if possible put a griffon in there to negate the terror, and use weird enchantment as well if you can for added punch. Of course it's almost always going to be at least a three order manouvre, and if you don't pull it off you can most likely say goodbye to that brigade the next turn as it will be HE's primary target for said dragon rider, etc...