Here is an excerpt from a dedicated page at MITT Doctoral school describing the procedure to apply and register for a thesis.
To apply for doctoral studies you must have a supervisor holding an HdR, member of the EDMITT, a topic and funding for the duration of the thesis.
The EDMITT does not put students in contact with researchers. You must consult the web pages of the different host research units to get in touch with a supervisor in the specialty in which you wish to work.
All applications are examined by one of the members of the ED committee in order to ensure that all the conditions are met for the thesis to proceed correctly.
This means receiving a salary in order to be able to devote yourself to doctoral research. You have to apply to various granting institutions. Sometimes there is a possibility or obligation to teach in addition to the research.
A distinction is made between initial funding, generally for 3 years, and complementary funding, which takes place after the 3 years, generally on one-year or 6-month A.T.E.R. (RTA) positions.
Initial funding (sample of possibilities) :
To find a thesis advisor you may (non-exhaustive list)
Study for a year in Master 2
Contact the Toulouse Mathematics Insititute (IMT)
Contact researchers directly
Everything is explained on the Application-Registration page of the MITT Doctoral School.