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Milk Bars in Warsaw's Old Town [18]
I wasn't sure whether they were comparable to a soup kitchen in the US in terms of the people who can be found there.
No, some are popular with office workers for lunch, with students, with retired people. Sometimes they also sell the food in plastic tubs to take out and reheat - elderly people often buy it. Some take food stamps, however the people with those tend to go to the suburban ones rather than the city centre.
Some are modern ones (the Mleczarnia chain in Warsaw, bright, shiny and clean) some are older ones that have been modernised, some are still very old fashioned looking (don't let that put you off) plus there are a few office canteens that are open to the public - very similar to Bary Mleczny and often better.
The soups are usually excellent in them.
Another option for eating is the catering college on ul. Poznanska. (the southern end) They have a daily menu, and always a set menu (a lot of the customers have a subscription and pay in advance to use a fixed menu - you can have the daily set price meal or order from a menu). It's Bar Mleczny prices more or less and very good quality.