Learn Hardcore Russian: В большой кастрюле варится борщ.

The suffix -ся on варится converts the active verb варить (“to cook/boil”) into a passive/reflexive meaning “is cooking” or “is being cooked.” This way, борщ becomes the grammatical subject, and the person doing the cooking is omitted.

Who is cooking the borscht? Why isn’t there an agent like они (they) or мы (we)?

Using the passive/reflexive form варится intentionally leaves the agent unspecified. The focus is on the process—борщ cooking in the pot—rather than on who is doing the cooking.

Why is кастрюле in this form? What case is it?

Кастрюле is in the prepositional case. After the preposition в, Russian uses:

accusative for motion into something,
prepositional for location inside something. Here it means “in the big pot,” so кастрюля takes the prepositional ending -е.

Why does большой end in -ой rather than -ая?

Adjectives agree with their nouns in gender, number, and case. Ка́стрюля is feminine, singular, and in the prepositional case. The correct adjective ending for that form is -ой, giving большой кастрюле.

What case is борщ, and why isn’t it in the accusative if it’s being cooked?

Борщ is in the nominative case, serving as the subject of the passive/reflexive verb варится. In Russian passive constructions with -ся, the thing undergoing the action (the “object” in English) becomes the nominative subject.

Could I say борщ варится в большой кастрюле instead? Does word order matter?

Yes. Russian word order is flexible:

В большой кастрюле варится борщ emphasizes location.
Борщ варится в большой кастрюле emphasizes борщ as the topic. Both are grammatically correct.

Can I substitute варится with готовится or кипит here?

Готовится means “is being prepared” (general cooking, not specifically boiling).
Кипит means “is boiling” (focus on reaching boiling point).
Use варится when you want to stress the actual boiling/cooking process of a dish like soup.

What aspect is варится, imperfective or perfective? When would I use the perfective?

Варится is imperfective, describing an ongoing process (“is cooking”). The perfective counterpart is сварится, which you’d use to express completion or future result: e.g., «Борщ сварится к обеду» (“The borscht will be cooked by lunchtime.”).