Learn Finnish: Kukistettuaan lohikäärmeen sankari löysi portaalin. - After defeating the dragon, the hero found the portal.

The base verb is kukistaa (to defeat). For the past temporal construction ("after doing something"), you start with the passive past participle kukistettu (defeated). Then you add the partitive ending -a to get kukistettua-. Finally, you attach the possessive suffix -an, which matches the main subject sankari. The formula is: passive past participle + partitive + possessive suffix.

Could I just use a kun (when) clause instead of kukistettuaan?

Yes, you could say Kun sankari oli kukistanut lohikäärmeen, hän löysi portaalin (When the hero had defeated the dragon, he found the portal). The meaning is exactly the same. However, the past temporal construction is a stylistic choice that makes the sentence more concise. It is very common in written Finnish, literature, and news.

Why is lohikäärmeen in the genitive case?

Even though it is attached to a participle construction instead of a regular main verb, lohikäärmeen is still a direct object. Because the action of defeating the dragon is completely finished, it takes the total object case, which for singular nouns looks identical to the genitive (the -n ending).

How do we know who defeated the dragon, since -an just means "his/hers/theirs"?