Maricopa County Courthouse says that the building was completed 1928-29 and was used as both the County Courthouse and as the Phoenix City Hall. It goes on to say 'The County Courthouse portion of the building was designed by Louisiana architect, Edward F Neild. The City chose the Phoenix firm of Lescher and Mahoney to construct the city hall'.
Waymaking.com have it the other way around and have spelt the architect's name as 'Nield'.
I would hope that the plaque on the building is correct but stranger things have happened.