Learn Finnish: Hän teeskenteli, mutta näin virneen hänen huomaamattaan. - She pretended, but I saw the smirk without her noticing.

It starts with the verb huomata (to notice). To say "without noticing", you use the 3rd infinitive abessive form: huomaamatta. To specify who isn't noticing, you add the 3rd-person possessive suffix -an to the end, which pairs with the genitive pronoun hänen (her/his).

Could we drop the word hänen and just say mutta näin virneen huomaamattaan?

No, hänen is mandatory here. In Finnish, you can only drop the pronoun if the person doing the unperformed action is the same as the subject of the main clause. Here, the main subject is "I" (näin), but the person not noticing is "she". Because the subjects are different, you must include hänen.

Is there a more basic way to say "without her noticing" if I can't remember the -mattaan structure on the fly?

Yes! You can always use ilman, että (without [that]) followed by a normal conjugated verb. In this case, you would say ilman, että hän huomasi. The -matta infinitive structure is just a more compact and elegant way to express the exact same thing.

Why is the word for "smirk" (virne) written with an -en ending as virneen?