While we're wishcasting, I wouldn't mind seeing something like Minneapolis' Midtown Global Market (as reported on by VI here), which might be characterized as a public market with an (multi-)ethnic focus, in Chicago. I visited last Labor Day weekend when in town for a wedding and thought it was outstanding.

Some thoughts on the French farmers' markets, which the OP mentioned. You'll note that at a Parisian market in December, many of the stands are selling meat and cheese, as wells as baked goods and cooked food: pizza, roast chickens, sausages, etc.. Others aren't selling any food at all, but offer clothes, jewelry, and so on -- in that sense, the "French Market" organized in Lakeview created by Bensidoun (the same company that organizes markets in Paris) is quite authentic.

Furthermore, many French market stands sell items not made by the stand owner. This isn't necessarily bad at all; often a cheesemaker or mushroom farmer will also sell his neighbor's honey, a friend's cured sausage, and someone else's special pudding. But it highlights the fact that many "farm stands" in Europe and the U.S. are less like the farmer bringing some corn and eggs to town and more like a specialty food store that only appears once a week. Many of the vegetable and fruit stands at Green City Market probably wouldn't find it practical or worthwhile to set up in Chicago year-round for a day or two a week. But some of the bakeries, butchers, and cheese-makers might.