That customer — you know the one — complained again, you know.
Literal
That [of-の] customer [topic-は], again grumble [object-を] came-to-pick [informative-わよ].
いちゃもんをつける ('to make complaints, pick a fight, grumble, be picky') is a casual/rough idiom. The speaker + つけてきた ('came (to) attach') = 'came at me with complaints.' The sentence-final わよ is the feminine assertion combination (わ adds a soft/feminine tone, よ is informative). Typical of gossipy spoken register.