Boudot was born in Morteau, in Franche Comté, in 1571.
He graduated doctor of the Sorbonne in 1604, and was appointed the episcopal official of Jean Richardot, bishop of Arras, following him to Cambrai as archdeacon when Richardot became archbishop there.
He was also appointed a preacher in ordinary to Albert VII, Archduke of Austria, preaching the funeral sermon for Albert's brother Emperor Rudolph II in the court chapel in Brussels in 1612.
In 1619 he was appointed bishop of Saint-Omer in place of Jacques Blaseus, O.F.M.
On 17 March of that year he gave his approval to and graced with indulgences an anonymous Marian devotional text, The Devotion of Bondage.