We feel confident that the M.E. initials on this plate stand for Methodist Episcopal, which was the first Methodist denomination in the United States. It could have been made for any number of churches. According to the Wikipedia, two branches of the church merged in 1939 to form the Methodist Church, so this plate, made in 1936, was soon to be obsolete!

As a point of interest, the Grafton church famous for the first Mother's Day observance was Andrews Methodist Episcopal. But considering the close proximity to the Carr plant, it would seem likely that any china made for that particular church would include the Andrews name as well.