Dessert needing Chartreuse — lots — sloshed (9,5)
I believe the answer is:
charlotte russe
'dessert' is the definition.
(charlotte russe is a kind of dessert)
'chartreuse lots sloshed' is the wordplay.
'sloshed' indicates an anagram.
'chartreuse'+'lots'='chartreuselots'
'chartreuselots' with letters rearranged gives 'CHARLOTTE RUSSE'.
'needing' is the link.
(Other definitions for charlotte russe that I've seen before include "Sponge/custard pudding" , "dessert" , "Lady's fingers cupping cream" , "Sponge and custard pudding" , "afters?" .)