Il cuoco mi ha spiegato che avrebbe infornato la melanzana e il peperone dieci minuti dopo.

In normal Italian, a singular countable noun like cuoco usually needs an article when it refers to a specific person: il cuoco = the cook.

So Il cuoco mi ha spiegato... means The cook explained to me...

If you said only cuoco here, it would sound incomplete in ordinary prose. Italian uses articles more regularly than English does.

Mi is the indirect object pronoun meaning to me.

So:

• Il cuoco mi ha spiegato = The cook explained to me
• Il cuoco ha spiegato a me is also possible, but it is more emphatic

Usually Italian prefers the short pronoun before the verb:

• mi ha spiegato
• ti ha spiegato
• gli ha spiegato

You would use a me especially for contrast or emphasis, as in Ha spiegato a me, non a te.

Ha spiegato is the passato prossimo of spiegare. It presents the explanation as a completed event in the past.

So the sentence is talking about one finished action: the cook explained something to me.

If you used spiegava, that would sound more like background description, an ongoing action, or repeated behavior:

• mi ha spiegato = he explained to me once / as a completed event
• mi spiegava = he was explaining to me / he used to explain to me

Here ha spiegato is the natural choice because the explanation is a single completed act.

Che introduces the clause containing what was explained. It means that.

• mi ha spiegato che... = he explained to me that...

In English, that is often optional:

• He explained to me that he would...
• He explained to me he would...

In Italian, che is normally kept. Leaving it out would usually sound wrong.

This is one of the most important points in the sentence.

Avrebbe infornato is the condizionale passato of infornare, but here it is being used to express the future in the past.

The idea is:

1. At some moment in the past, the cook explained something.
2. From that past point of view, the oven action was still in the future.

A direct version might have been:

• Infornerò la melanzana e il peperone fra dieci minuti.

When this gets reported from a past point of view, Italian often changes it to:

• ha spiegato che avrebbe infornato... dieci minuti dopo

So avrebbe infornato here does not mainly mean a hypothetical would have put in the oven. It means something like would later put in the oven or was going to put in the oven.

Not exactly.

In English, would have put in the oven often sounds hypothetical or contrary to fact:

• He would have put it in the oven, but he forgot.

But in this Italian sentence, avrebbe infornato is more naturally understood as future from a past viewpoint:

• he would put it in the oven ten minutes later
• he was going to put it in the oven ten minutes later

So the Italian form is a conditional perfect, but the function in this sentence is not really hypothetical. It is temporal: future relative to a past moment.

Because spiegare usually introduces information presented as factual or at least as content being reported, not as doubt, emotion, or uncertainty.

So after ha spiegato che, Italian normally uses the indicative system, and here that includes the future in the past structure:

• ha spiegato che avrebbe infornato...

A subjunctive would not be the normal choice here.

Very roughly:

• verbs of saying/explaining/reporting often take ordinary statement forms
• verbs expressing doubt, fear, desire, opinion, etc. more often trigger the subjunctive in the right contexts

Infornare literally means to put into the oven.

That is slightly different from simply to bake in English, because infornare focuses on the moment of placing something in the oven, not necessarily the whole cooking process.

So:

• infornare = to put in the oven
• cuocere al forno = to bake / cook in the oven

In many real contexts, English may still translate it as bake, but the Italian verb itself specifically points to the oven-loading action.

Because the sentence is referring to one eggplant and one pepper.

Italian often uses singular count nouns with articles when talking about specific items:

• la melanzana = the eggplant
• il peperone = the pepper

If it meant multiple eggplants and peppers, you would expect plurals:

• le melanzane e i peperoni

So the singular forms suggest one of each.

Because both nouns are definite and singular, and in Italian each one normally keeps its own article.

So:

• la melanzana e il peperone

is the normal way to say:

• the eggplant and the pepper

Leaving out the second article would usually sound unnatural here.

Italian is often more explicit than English with articles in coordinated noun phrases.

It means ten minutes later or ten minutes afterwards.

It places the oven action at a point ten minutes after the reference moment. In this sentence, that reference moment is the past situation from which the cook was speaking/explaining.

So the timeline is:

• first: the cook explained
• then: ten minutes later, he would put the vegetables in the oven

This fits well with the future in the past idea created by avrebbe infornato.

Yes, though in many contexts they are close.

• dieci minuti dopo = ten minutes later
• dopo dieci minuti = after ten minutes

The first often sounds more like a point later on a timeline.
The second can sound more like after waiting ten minutes or once ten minutes had passed.

In this sentence, dieci minuti dopo works especially well because it matches reported speech and a past reference point.

Because with avere, the past participle usually does not agree with the direct object when that object comes after the verb.

So Italian says:

• ha mangiato la mela
• ha mangiato le mele
• ha infornato la melanzana e il peperone

In all of these, the participle stays in its basic form.

Agreement is more likely when the direct object is a pronoun placed before the verb:

• Le ha mangiate
• Le ha infornate

But in your sentence, the objects come after the verb, so infornato stays unchanged.