Learn Finnish: Heti kun olemme pilkkoneet sipulin, lisäämme sen keittoon. - As soon as we have chopped the onion, we will add it to the soup.

This is how Finnish handles the sequence of tenses with time words like heti kun (as soon as) or kun (when). To show that a future action must be completely finished before the next one starts, the completed action takes the perfect tense. The main action then takes the present tense (lisäämme).

Could we use the present tense for both verbs, saying Heti kun pilkomme..., lisäämme...?

You could, but it changes the meaning slightly. Using the present tense for both implies the actions are happening simultaneously. The perfect-present sequence specifically emphasizes the strict chronological order: we must finish the chopping completely before the adding can begin.

Why are sipulin and sen in the genitive-like accusative case instead of the partitive?

Both are total objects. The perfect tense already emphasizes completion, meaning you finish chopping the entire onion (or a specific intended amount), and then you add that whole amount to the soup. If you used the partitive (sipulia, sitä), it would imply an incomplete ongoing action or an undefined amount.

Why is the soup in the illative case (keittoon)?

This is due to verb rection. The verb lisätä (to add) requires the destination to be in the illative case (the 'into' case). In Finnish logic, you are moving the ingredient into the soup, so the word takes the -on ending here.