Egghead goes under in charge - that's cool (3)
I believe the answer is:
ice
'cool' is the definition.
(I know that ice is a more specific form of the action cool)
'egghead goes under in charge' is the wordplay.
'egghead' becomes 'e' (head letter of 'egg').
'goes under' is a charade indicator (letters next to each other).
'in charge' becomes 'ic' (abbreviation for 'in charge').
'e' put after 'ic' is 'ICE'.
'that's' is the link.
(Other definitions for ice that I've seen before include "Constituent of 2dn" , "Cooler" , "To put in a drink or slide on" , "Frozen liquid" , "Diamonds (informal)" .)