From LMA: Founded January 1854 in Fashion Street with the aim to supply soup, bread and meat twice a week, during the winter to the Jewish poor. In 1902 it moved to 17-19 Butler Street, renamed Brune Street in 1937. The building looks to have been purpose built, but only part of it was used for the charity work. The main source of funds was from the rental of the rest of the property.
Until WW2 it cooked soup in its own kitchen for consumption on the premises or to be taken away in tin cans. Rationing brought that to an end; bread and groceries were dispensed instead and what had been the kitchen space was rented out. It closed in 1992 and the services were merged with those at Jewish Care in Beaumont Grove. The building in Brune Street was converted to flats.
Some websites say that this was originally based in Leman Street but PortCities says that was the Poor Jews' Shelter.
Spitalfields Life have a whole post about The Soup Kitchens Of Spitalfields, including this image, which is captioned "The Jewish Soup Kitchen in Spitalfields, 1879 (Illustrated London News)".