Paul s’est plaint parce qu’il n’y avait plus de brioche au petit-déjeuner.

The verb is se plaindre, which means to complain. In French, this is a pronominal verb, so it normally comes with a reflexive pronoun: je me plains, tu te plains, il se plaint.

In the passé composé, pronominal verbs use être, so:

• il se plaint = he complains
• il s’est plaint = he complained

Even though it looks reflexive, you usually translate it simply as to complain, not to complain oneself.

Yes, it is irregular. The infinitive is plaindre, and its past participle is plaint.

So:

• plaindre → plaint
• se plaindre → s’est plaint

French has several verbs like this, for example:

• craindre → craint
• joindre → joint

So this is something you mainly have to learn as a verb form.

French often uses:

• passé composé for a completed event
• imparfait for background, description, or an ongoing situation

Here, Paul s’est plaint is the main event: he complained.
Il n’y avait plus de brioche gives the background situation that explains why he complained.

So the pattern is very natural:

• completed action: he complained
• background circumstance: there was no more brioche

No. In il y avait, the il is impersonal. It does not mean he, and it does not refer to Paul.

Il y a is the fixed French expression for there is / there are.
So:

• il y a = there is / there are
• il y avait = there was / there were

So qu’il n’y avait plus de brioche means that there was no more brioche, not that he had no more brioche.

The y is part of the fixed expression il y a. You should learn il y a as a whole unit meaning there is / there are.

Then you can change the tense:

• il y a = there is / there are
• il y avait = there was / there were
• il y aura = there will be

So in this sentence, y is not being used in some separate, literal way that you need to translate word for word. It is part of the standard existential expression.

Ne ... plus means no longer or not anymore.

So:

• il y avait encore de la brioche = there was still some brioche
• il n’y avait plus de brioche = there was no more brioche / there wasn’t any brioche left

In a simple tense like the imperfect, the negative structure goes around the conjugated verb:

• n’ before the verb group
• plus after the conjugated verb

That is why you get il n’y avait plus.

After negation, French often changes du / de la / des to just de.

Compare:

• Il y avait de la brioche. = There was some brioche.
• Il n’y avait plus de brioche. = There was no more brioche.

So plus de brioche is the normal form here. It means there was no brioche left anymore.

Because French usually avoids que directly before a vowel sound. The e drops, and you get an apostrophe:

• que il → qu’il
• que elle → qu’elle

This is called elision.

The same thing happens elsewhere in the sentence:

• se est → s’est
• ne y avait → n’y avait

French does this a lot to make pronunciation smoother.

Au is the contraction of à + le.

So:

• à le → au

Here, au petit-déjeuner means at breakfast or for breakfast, depending on how you phrase it in English.

It refers to the breakfast meal/time. So the sentence is saying that there was no more brioche available at breakfast.

Yes. Se plaindre de is very common when you say what someone is complaining about.

For example:

• Paul s’est plaint du bruit. = Paul complained about the noise.
• Paul s’est plaint du manque de brioche. = Paul complained about the lack of brioche.

In your sentence, French uses parce que to give the reason directly: he complained because there was no more brioche. That is completely natural.

In ne ... plus, the final s is often not pronounced. So it is usually said like plu.

That helps distinguish it from affirmative plus meaning more, where the s is often pronounced.

So:

• plus in il n’y avait plus de brioche ≈ plu
• plus meaning more is often pronounced with the s

For a learner, the main thing is: here plus is negative and means no more / not anymore.