If it's not much of a milling machine, avoid solid carbide. The slop in the machine will cause you to chip the cutter in no time. Unfortunately, HSS tools tend to be cheap junk or somewhat low tech these days.
rx8pilot says to change the design, and I'm with him on that. I sort of bridge the gap: sometimes hammering something out on a sloppy cheap manual machine or sometimes drawing for CNC or manual. When I first did drawings, the manual experience I had helped but I would still put my contact info right on the drawing and meet with the machinist to look over my drawings to find trouble spots. I would have appreciated a call that said "this deep thin slot is gonna cost ya!"
I've done 1" deep with a 1/4" endmill like the one Lightages points you towards in aluminium (I'm talkin 6061-T6 which is dead easy) without much trouble. It was slow since tool deflection is a real serious problem. 2" makes me shudder, even with a 3/8". If you can mill from both sides (like going through) I would do that if at all possible. Chip evacuation from 2" down itself going to be some work.