A 29-year-old man was assaulted by another man inside Tasty Bites on Paisley Road West, near Tradeston, at about 22:40 on Sunday 17 April.
The victim was treated in hospital and has been left with facial scarring.
Police said there were a number of people in the shop at the time and the attacker was not known to the victim.
The man police want to trace is described as white, with short cropped dark hair.
He spoke with a Glaswegian accent and at the time of the attack was wearing a cream cardigan and dark jeans.
Det Con Nikki McPherson said: "This was a brutal attack that left a man seriously injured and scarred for life. It is important that we trace the person responsible for this reckless behaviour as soon as possible.
"We have issued an image of a man who we wish to speak to in connection with this incident and I would ask anyone who recognises him to get in touch.
"I would also appeal to anyone who witnessed this incident and is yet to come forward and speak to police, to contact officers."
Police have issued a CCTV image of a man they want to trace in connection with a "brutal attack" at a fast food takeaway shop in Glasgow.