You will note that both Doxygen and Sphinx are run, despite the fact that only the invocation to Sphinx was given when declaring the custom BREATHE_DOC_TARGET_NAME target. This is thanks to the following settings, defined in the conf.py file for Sphinx:

def run_doxygen(folder):
    """Run the doxygen make command in the designated folder"""

    try:
        retcode = subprocess.call("cd {}; doxygen".format(folder), shell=True)
        if retcode < 0:
            sys.stderr.write(
                "doxygen terminated by signal {}".format(-retcode))
    except OSError as e:
        sys.stderr.write("doxygen execution failed: {}".format(e))


def setup(app):
    run_doxygen('@BREATHE_DOC_BUILD_DIR@')

Doxygen will generate XML output that the Breathe plugin will be able to render in a form consistent with the chosen Sphinx documentation style.