Yiannis is good in Lindos - particularly if you like a bit of people watching :D!
Apparently Billy's Bar in Lindos is good if you like a proper english breakfast but i don't so it's just what i've heard.

Go to the Inn on the Road in Kalathos - Nico & Despina excellent hosts - have gone there for breakfast every day of our last 6 trips to Kalathos/Pekfos/Lardos - great hosts - just back yesterday - missing it already

There's only one place to go for a breakfast in Lindos and that's at the
apollo right opposite the bakers where you start to walk up to the acropolis betty who does the breakfasts lived in minnesotta for a while
you'll hear her american accent all over the bar.The breakfasts are huge
with everything on them hash browns the lot.Also try one of Georges
(her husband)shorts on the night they are absolutely lethal.
Rick.

totally agree - The Apollo