Je suis si fier d’elle pour avoir mené sa thèse à bien.

Without including I am ? :

Tellement fier d’elle pour avoir terminé sa thèse!

Please note that you should write fière if you are a she.
And… in no case fiert.

Please note that this would be somehow colloquial.

A bit less colloquial but… BTW…less common :

• Tellement fier d’elle qui a terminé cette thèse.
• Tellement fier d’elle à propos de l’achèvement de sa thèse.

I’m afraid there is nothing in French for a perfect translation of that verbless sentence. Traditionnally, you do not use verbless sentences in French. “Tellement fier d’elle après qu’elle a eu fini sa thèse!” and the like of it has not yet been introduced into the language and you would not give the impression of speaking correctly. It is possible to do what you wish in writing but in speech you have to handle the language in such a skilful manner, have developped such a talent that it is better to forget about it; very few French speakers have the necessary poise to allow themselves such uses of the language. So, in writing you could say, if properly introduced by what precedes, something like this below;

• Quelle fierté que la mienne à la nouvelle qu’elle avait fini sa thèse!

I advise you to stick to a normal French sentence in which there is a verb, however, if you don’t, you’ll be understood anyway.