That is to say what is the oldest record we have of what you or I would call a restaurant. For this example let's say: A perminate structure with reasonable hours (e.g. not only serving 3 meals a day) where you could enter, choose a drink and an entre from a menu and eat with your family before leaving.
I would say that two details might be narrowing the answer more than you think. Written menus were probably not common before the 18th century, because writing surfaces were expensive and not everyone could read and it was easier to have the server just explain what was available (I think there is one Roman inscription with a price list, but it is on stone). So if you visited a Roman taberna, they would probably have two or three grades each of wine and bread and oil, and maybe two or three hot dishes simmering on a griddle, and the proprietor would tell you if you could not see and answer your questions about prices. Maybe his did not have something you wanted as part of the meal, but the taberna two doors down did.
Second, the idea of a sit-down restaurant as a place for the polite family is not so common. You could go have something to eat with your beer at an Old Babylonian beer house 4000 years ago, but they were full of drunks and disreputable people and women of negotiable virtue. This gets us into the comment by disgruntledape about how some aspects of American restaurant culture are very recent marketing choices (and others go back to the aftermath of the French Revolution when aristocrats' personal chefs needed to find a way of making a living without a single patron) whereas the basic idea of a restaurant goes back thousands of years.
Do you mean Modern as in multiple courses cooked to order? Or are you referring to Modern as the dishes being prepared?