What is "followed by a suitably chilled eating establishment"? In other words, please?

Context is "We wandered along the river, enjoying a gentle 鈥榟air of the dog鈥?glass of red followed by a suitably chilled eating establishment."

"Hair of the dog" refers to having a drink in order to dispel the effects of what alcohol had done previously. The idea of a RED is a glass of red wine, which is usually consumed at room temperature on the cool side. A suitably chilled eating establishment would be a restaurant which is also chilled on the cool side of room temperature.

That's a little mysterious. Without any other context, I would say it means a restaurant that is suitably cool, in temperature?

I'm going to guess that the restaurant's air conditioning was to the writer's liking.

A suitably chilled eating establishment? That makes no sense. I never heard anybody say that.

It means "an air-conditioned restaurant".