Grøntsagerne ryger i en stor gryde, og jeg tænder for komfuret.

Grøntsagerne is definite plural: the vegetables.

• en grøntsag = a vegetable
• grøntsager = vegetables (indefinite plural)
• grøntsagerne = the vegetables (definite plural)

The ending -ne is a common definite plural ending in Danish.

Yes. Danish often uses present tense for actions happening now or as instructions in a recipe-style narrative.
ryger is present of at ryge. In this sentence it functions like “go/throw/go in” (see next question), and the present tense makes it feel immediate: “The vegetables go into a big pot…”

At ryge can mean “to smoke,” but it’s also used idiomatically for something that gets tossed/ends up somewhere quickly, especially in cooking or casual speech.
So Grøntsagerne ryger i en stor gryde is like: “The vegetables go/are tossed into a big pot.”

It’s informal and vivid; it doesn’t imply actual smoke here.

It’s not grammatically passive. Danish passive is typically -s (e.g., puttes) or bliver + past participle.
Here it’s an active intransitive use: grøntsagerne are the grammatical subject, and ryger describes what happens to them (“they go/are tossed”).

If you wanted a clearer “someone puts them in,” you might see something like Jeg putter grøntsagerne i en stor gryde.

Because i means in/into and is used when something ends up inside a container. A pot is a container, so i en gryde is natural.

• i = in/into (inside)
• på = on (on a surface)
• til = to (toward a destination, not necessarily “inside”)

Also, Danish often uses i with verbs of movement in this kind of “into” sense.

Adjectives usually come before the noun in Danish, like in English.
Here the noun is common gender (en gryde), singular, indefinite, so the adjective is the basic form: stor.

Compare:

• en stor gryde (a big pot)
• gryden er stor (the pot is big)
• et stort hus (a big house — neuter et, so stort)
• de store gryder (the big pots — plural store)

In Danish, the common expression for switching on a device/appliance is at tænde for + something:

• tænde for komfuret = turn on the stove
• tænde for lyset = turn on the light

You can sometimes see tænde + noun (especially with fire/cigarettes, etc.), but for appliances tænde for is the standard pattern.

They’re opposites for switching devices on/off:

• at tænde (for) = to turn on
• at slukke (for) = to turn off

Example: Jeg slukker for komfuret = I turn off the stove.

Komfur is neuter gender (et komfur). The definite singular is formed with -et:

• et komfur = a stove/cooker
• komfuret = the stove/cooker

(For common gender nouns you often see -en, e.g., gryden = the pot.)

Yes. Danish typically uses a comma to separate two independent clauses (each could stand as its own sentence): 1) Grøntsagerne ryger i en stor gryde
2) jeg tænder for komfuret

Because both parts have their own verb (ryger, tænder), the comma before og is normal.

After og (and other coordinating conjunctions), Danish keeps normal main-clause word order: subject before verb, as in jeg tænder.

In contrast, after many subordinating conjunctions (like fordi, at, når), Danish typically sends the verb later (verb-final-ish), e.g. fordi jeg tænder for komfuret.