Greenville's most impressive breakfast possibly is in N Eastern St (7)

I haven't cracked this one yet — but the brilliant team at Times for the Times have:

EAT (to breakfast) in N E ST; Greenville is a city in South Carolina; allegedly, then, denizens of the good ol’ US of A use the word ‘neat’ to  mean ‘good’. All I can say is that, when I visited my mother’s family in New Zealand for the first time in 1977, everyone was saying things were ‘real neat-aye,’ with an alarmingly ascending intonation.