Después de que la veterinaria la viera, mi tortuga volvió a comer pienso.

Because después de que is the usual structure when after is followed by a whole clause with its own conjugated verb.

Here, the clause is la veterinaria la viera.

Compare:

• Después de que la veterinaria la vio/viera... = after the vet saw her...
• Después de la consulta... = after the appointment...
• Después de comer... = after eating...

So después by itself usually means afterwards, while después de que introduces after + subject + verb.

Viera is the imperfect subjunctive of ver.

So:

• ver = to see
• viera = saw / were to see / might see, depending on context

In this sentence, it refers to the vet seeing the tortoise.

This is the part many learners notice first.

The basic rule is:

• después de que + indicative for events presented as real and completed
• después de que + subjunctive when the action is future, pending, or viewed from another reference point

So for a straightforward completed past event, many learners would expect:

Después de que la veterinaria la vio, mi tortuga volvió a comer pienso.

That is probably the most neutral version for everyday learning.

The version with viera is understandable, and you may come across similar uses, but for a simple completed past event, vio is the safer default to learn first.

Yes. In fact, Después de que la veterinaria la vio, mi tortuga volvió a comer pienso is a very natural choice.

A useful contrast is:

• Te llamaré después de que la veterinaria la vea.
  Future idea, so subjunctive
• Mi tortuga volvió a comer después de que la veterinaria la vio.
  Completed past event, so indicative

So if you are unsure, vio is the more straightforward option here.

It is a direct object pronoun, and it refers to mi tortuga.

So:

• la veterinaria = the vet
• la viera = saw her/it
• mi tortuga = the thing being referred to

English would say after the vet saw her, and Spanish does the same with la.

Spanish allows this. You can read the sentence and only later discover exactly who la refers to.

Once you reach mi tortuga, it becomes clear that la means her/it, referring to the tortoise.

This can feel slightly unusual to English speakers, because English often prefers to mention the noun first. But in Spanish, this kind of reference is normal in longer sentences.

Here it almost certainly means the female vet.

As a noun, veterinaria can mean:

• female veterinarian
• sometimes, in other contexts, veterinary clinic

But in this sentence, the subject is someone who saw the tortoise, so the most natural reading is the female vet.

Volver a + infinitive is a very common Spanish structure meaning to do something again or to resume doing something.

So:

• volvió a comer = started eating again / resumed eating
• comió otra vez = ate again

In this sentence, volvió a comer suggests that the tortoise had stopped eating for a while and then began again. That is a little more natural than comió otra vez in this context.

Because the sentence presents the return to eating as a completed event.

Volvió marks the moment when the tortoise resumed eating.

If you used volvía, that would sound more like background description, an ongoing situation, or a habitual action in the past. Here, the speaker is talking about a specific change, so volvió fits well.

Yes, pienso can mean I think, but not here.

In this sentence, pienso is a noun meaning animal feed / pet food, especially in Spain. For a tortoise, it suggests prepared food or pellets.

So:

• pienso as a verb = I think
• pienso as a noun = feed / pet food

The context makes it clear that this is the noun.

Because Spanish often uses food nouns without an article after verbs like comer when speaking in a general sense.

For example:

• comer pan
• comer fruta
• comer arroz
• comer pienso

So comer pienso means to eat feed/pet food in a general way. If you said el pienso, that would sound more specific, as if you meant a particular feed.

Yes. You can also say:

Mi tortuga volvió a comer pienso después de que la veterinaria la vio.

That means the same thing. The difference is mostly one of emphasis:

• Después de que..., main clause = sets the time frame first
• main clause + después de que... = gives the main event first

Both are natural.

Not with the same meaning.

Después de + infinitive is normally used when the subject is the same in both parts.

For example:

• Después de comer, mi tortuga se durmió.
  The tortoise is the one doing both actions.

But in your sentence, the subject of the first action is la veterinaria, while the subject of the second is mi tortuga. Because the subjects are different, Spanish uses después de que + verb instead.