Learn Hardcore Finnish: Tänä vuonna piparkakut on leivottu ajoissa, joten voimme vain nauttia niistä. - This year the gingerbread cookies have been baked on time, so we can just enjoy them.

In piparkakut on leivottu, the verb phrase on leivottu is the impersonal passive perfect.
It literally means “has been baked” / “have been baked (by someone)”, with no subject.

on leivottu = passive perfect (“has been baked”)
piparkakut = the thing that has been baked (the object, not the subject)

In Finnish, the passive has only one form; it does not agree with number.
Using ovat leivottu would try to combine the personal plural (ovat) with a passive participle (leivottu), which is grammatically wrong in standard Finnish.

So what exactly does on leivottu mean, and how is it formed?

On leivottu is the present perfect passive of leipoa (“to bake”):

on = 3rd person singular of olla (“to be / to have”) used as an auxiliary
leivottu = passive past participle of leipoa

Together: on leivottu ≈ “has been baked” / “have been baked (by someone)”.

Compare:

leivotaan – “(they) bake / is baked” (present passive)
leivottiin – “(they) baked / was baked” (past passive)
on leivottu – “(they) have baked / has been baked” (perfect passive)

How is on leivottu different from leivottiin?

Both are passive, but they refer to different tenses:

piparkakut leivottiin ajoissa
= “the gingerbread cookies were baked on time” (simple past – the action is just in the past)
piparkakut on leivottu ajoissa
= “the gingerbread cookies have been baked on time” (present perfect – the result is relevant now, they are already baked at this point)

The sentence you gave uses the perfect because it connects naturally to so now we can enjoy them.

Why is it Tänä vuonna, and what are tänä and vuonna grammatically?

Tänä vuonna literally means “in this year”, used as “this year”.

tänä = essive singular of tämä (“this”)
vuonna = essive singular of vuosi (“year”)

So grammatically it’s “as this year” / “in the role of this year”.
Finnish often uses essive (-na/-nä) for time expressions:

tänä aamuna – this morning
tänä kesänä – this summer

Saying something like tässä vuodessa is not used for “this year” in the time sense.

Why is it niistä and not ne or niitä?

Niistä is the elative plural form of ne (“they / those”), and it’s required by the verb nauttia.

nauttia jostakin = “to enjoy something / take pleasure in something”
→ “jostakin” is in the elative (-sta / -stä)
niistä = “from them / of them / in them” (here: “enjoy them”)

So:

nauttia niistä = “enjoy them / take pleasure in them”

Using:

ne would be nominative and ungrammatical here.
niitä would be partitive and doesn’t fit the verb pattern nauttia + elative.

Could we say nauttia piparkakkuja instead of nauttia niistä?

Yes, but the nuance changes a bit:

nauttia piparkak(u)ja (partitive plural)
focuses more on consuming the cookies (eating/drinking).
nauttia niistä (elative)
follows the pattern nauttia jostakin and is more about enjoying them in general (their existence, taste, smell, the situation, etc.).

In your sentence, nauttia niistä feels like “just relax and enjoy (the fact that) they’re there / we have them”.

What does vain do in voimme vain nauttia niistä, and could its position change?

Vain means “only / just”. In voimme vain nauttia niistä it modifies the verb phrase “nauttia niistä”:

voimme vain nauttia niistä
≈ “we can just enjoy them” / “all we need to do is enjoy them”

If you move vain, the meaning changes:

voimme nauttia vain niistä
≈ “we can enjoy only them (and not anything else)”

So:

vain before the verb → “only this action”
vain before the object/pronoun → “only this thing/person”

Why is it just voimme and not me voimme?

Finnish verb forms already show the person, so the pronoun is often dropped:

voimme = “we can” (the -mme ending marks 1st person plural)
me voimme is also correct, but me is only needed for emphasis or contrast:

Examples:

Voimme vain nauttia niistä. – “We can just enjoy them.” (neutral)
Me voimme vain nauttia niistä. – “We can just enjoy them (as opposed to others).”

What does joten mean here, and how is it used?

Joten means “so / therefore”, introducing a result or consequence clause.

In your sentence:

Tänä vuonna piparkakut on leivottu ajoissa, joten voimme vain nauttia niistä.
= “…, so we can just enjoy them.”

It is similar to:

siksi – “for that reason, that’s why” (usually adverb at the start: Siksi voimme…)
niin että – “so that” (more like purpose/result clause, often with its own nuances)

Joten directly connects two clauses, like English “so / therefore” between sentences.

What does ajoissa mean, and what form is it?

Ajoissa means “on time / in time”.

It’s an adverb derived from aika (“time”), but it’s used as a fixed adverbial form, not something you usually decline productively.
So:

piparkakut on leivottu ajoissa
= “the gingerbread cookies have been baked on time / in good time”

You might also see ajallaan with a similar meaning in some contexts, but ajoissa is very common for “on time”.

Can the word order change, for example to Piparkakut on tänä vuonna leivottu ajoissa?

Yes, word order in Finnish is relatively flexible, and Piparkakut on tänä vuonna leivottu ajoissa is grammatically fine.

The differences are subtle:

Tänä vuonna piparkakut on leivottu ajoissa…
→ Emphasis starts with this year as the contrast: this year (unlike other years), the cookies were baked on time…
Piparkakut on tänä vuonna leivottu ajoissa…
→ Starts with the cookies, then adds this year as extra information.

Both are natural; the original just foregrounds “this year” a bit more.