As a professor myself, your gratitude is more than enough. In some way, your great success is a bit of ours, and that is what makes our success great too: integration. I would not worry but a mail some time is very welcome indeed.
I am a civil servant (Spain), I cannot legally accept them.
But I have indeed been gifted a Christmas "basket" (with typical Christmas food) and from another student an amazon box with "serrano" produce. I could not refuse them (the first one because it was already brought to my office, the second one had no return address afaicr).
It was very very pleasing, of course. And neither was really expensive.
I think both students hit the exact level of compliment: singular but not extravagant.
ETA: thanks for the compliment.
This is a lofty sentiment, and I mean that in a complimentary way, but how might someone like you respond to a cash gift, if you can expand?