Paxboy
what i really do, when ATC wants me up quickly, is:
Let's say I'm climbing at 320 KIAS (indicated airspeed) at climb thrust, receiving the ATC request, I select 270 KIAS on the autopilot, maintaining the thrust as it is, the only way for the aircraft to catch up the 270 KIAS is to change the AOA(Angle Of Attack) and consequently the attitude (the airplane will point the nose more up); this way I obtain a very good rate of climb (in the order of 3500 to 4000 feet/minute: it depends on the aircraft's weight), will not last long (a bit longer if i use MCT) . When the energy has dissipated, the rate of climb will settle down at say 1500 feet/minute and 270KIAS, but ATC has seen its altitude readout spinning really fast!
We cannot always satisfy ATC; sometimes they want performance the aircraft can't make; we tell them: so they give us altitude restriction or heading to steer to momentarily.
hope this can help
best
B_D