Great food and a game of chess? Is it out there?
I not only can think of none in L.A. (the only place in LA I know where there's ongoing chess games is Tang's Donuts at Fountain and Sunset) I'm going out on a limb and opining that no such place exists in any major city in the U.S. The dictates of urban restaurant economics just weigh too hard against i;. American chess players, as a group, are not notoriously big spenders on food and wine. . .
I will tend to disagree with the notion that no American city could support it, since certainly LA could, especially between the hours of 2pm and 5pm when many restaurants opt to close out right.
I wouldn't look for specifically a good game of chess, but rather find a spot with good sidewalk dining, as chess, in my opinion, is a game that is best played close to the pavement. The hum and buzz of the world going by falls away as you start thinking of moves five, ten and fifteen steps ahead. (I'm sure many Angelenos share a vision of chess being played en masse in New York's Central Park.)
My immediate choice would be places along Beverly, especially, BLD, Milk (though no wine) and King's Road, but also along 3rd like Little Door, Toast, Quality and anyone else who has sidewalk seating. But really any neighborhood joint would do, but those are the ones where I find myself most often loitering reading a book alone.
And most definitely the Farmer's Market would allow it. Grab a glass of wine from Monsieur Marcel and you can play all the chess you want.