Nothing is known about Franz-Josef Müller’s training, though he was probably apprenticed to his father Michael IV Müller (1627–1682). He only became a member of the Brotherhood of St Luke (the painters’ guild) in 1697, even though he had taken over his father’s workshop in 1682. His output was extensive, and examples may be found in many public and private collections.
Among his larger series is the cycle of armorial panels for the Beat Zurlauben-Reding, Reding, and von Beroldingen families, some of which bear monogram signatures, of either IM (Josef Müller) or FIM (Franz-Josef Müller). There are further series in a chapel in Valais canton, and the ossuary at Unterschächen in Uri canton. The panel considered to be his most important work is one donated by the Abbot of Engelberg in 1688, which Müller created as part of a series for the convent in Altdorf (Uri canton).
For a detailed description of Müller’s life, see Uta Bergmann, Die Zuger Glasmalerei des 16. bis 18. Jahrhunderts, Corpus Vitrearum, Schweiz, Reihe Neuzeit, 4, 2004, pp. 122–25, with numerous illustrations.