This produces the apparently tautologous "What am I experiencing when... I experience myself thinking?" ("De quoi fais-je l'épreuve quand... je m'éprouve pensant?") but perhaps this can be worked around.

The contrast is between the WHAT that is the object of my thought (la pensée) - and the subjective side of thought - the experience of thought (l'expérience de la pensée).

This is expressed in the French in the difference between "faire l'épreuve de", where an experience is the basis of an objective judgement ("Action d'éprouver, opération à l'aide de laquelle on juge si une chose a la qualité que nous lui croyons") and "éprouver", where experience is just a subjective feeling.

Ie the first "experience" is the sense of something being "testified to", you could, eg. say "what is being testified to when I experience myself thinking", but I find that awkward and a bit off and it loses the subtlety that is being expressed: the duality of experience, its two 'sides', one inside the other.

I suggest enhancing the distinction by translating the "éprouver" as "feeling" - "What am I experiencing when I feel myself thinking?" - perhaps also italicising the "what" and trusting that the context makes the contrast clear... but it is tricky!