Uff! I don't know about the best paella in town....if I leave out my mother's out of the competition that is!!!
(and my own veggie one :o )
And, I confess I've only been to the famous (specially amongst 'guiris') Champagnería/Arroceria Gala
www.paellas-gala.com once (it happens when you live a couple of minutes from a place), and wasn't at all impressed by their food.
Of course we ordered some quite non-conventional dishes such as "fideua mixta", "paella oriental", etc... so I really can't tell you if they master the more classic stuff such as paella and arroz a banda....
Well, while paella is definitely NOT a Madrilenian dish, 'Callos a la madrileña' o 'Cocido Madrileño' are....
Personally I don't like callos, which are tripe :p , but see some ideas for cocido places on this recent post:
http://www.madridman.com/ubb/ultimatebb.php?ubb=get_topic;f=4;t=001049
Other dishes I wouldn't miss would be a nice Fabada, Gazpacho, Bacalao al pil-pil or a la Vizcaína, Pisto, Migas, Menestra de verduras, Pollo al ajillo or a la chilindrón, Caldereta....