This building clearly features on an 1894 map and, on the 1895 valuation roll, a Janet Grieve or Simpson (flesher) is listed as owner of three entries ('Slaughterhouse and Stable', 'House, Shop & Garden' and 'Huts') on Hillhead. Janet was by this time the widow of the butcher David Simpson who had died earlier that year after a fall from the Largo viaduct.
Looking at other valuation roll entries and at census information, it seems that David Simpson had moved to Largo parish around 1878 from Markinch. Initially, he was a flesher in Kirkton of Largo before moving to Lundin Mill Farm before 1885. By the 1891 census, Simpson has his own butcher's shop, which I'm guessing is the same one noted as belonging to his widow four years later on Hillhead. So, more than likely the slaughterhouse was built for him sometime between 1885 and 1891.