Were there any chain stores or restaurants during the Roman Empire

Whilst potentially not exactly what you had in mind, potters certainly set up workshops far away from their original location and produced pots bearing the same signature.

Whilst I cannot speak confidently for the entire empire, for Roman Britain it is certainly true that potters with workshops in Colchester set up additional workshops on the Antonine frontier to supply soldiers moving north. I can provide some citations this evening when I dig out my MPhil thesis, hope this helps! (And sorry if this answer breaks the rules)

Edit: Some sources as promised:

For an excellent discussion of pottery on the Antonine Wall, including the aforementioned workshops, see David Breeze's excavations at Bearsden 'Breeze, D., 2016. Bearsden: A Roman Fort on the Antonine Wall. Edinburgh: Society of Antiquaries Scotland'

The ScARF Report on Roman scotland gives an excellent overview and can be read online here :http://www.scottishheritagehub.com/sites/default/files/u12/ScARF%20Roman%20June%202012.pdf Particular attention paid to pg48, and to a strong bibliography

For a more general overview of the Antonine Wall finds with mention of famous potters marks see Robertson's guidebook, 'Robertson, A. S. & Keppie, L., 2001. The Antonine Wall: A Handbook to the surviving remains. Glasgow: Glasgow Archaeological Society'

I'm afraid most of the other pieces of evidence I've come across are fragmentary mentions in finds reports of notable British potters such as Sarrius, and so it would be too time consuming to dig them all out and post them here. Unfortunately my specialism is more the Antonine Wall than Roman pottery generally. Hope someone else can add to this and you get a good answer!