A mia sorella piacciono i broccoli al forno.

Because piacere does not work like English to like.

Italian structures it more like:

To my sister are pleasing oven-baked broccoli.

So:

• a mia sorella = to my sister
• piacciono = are pleasing
• i broccoli al forno = the oven-baked broccoli

With piacere, the person who experiences the liking is usually introduced by a.

Because the verb agrees with the thing being liked, not with the person.

Here, the thing being liked is:

i broccoli al forno

That is plural, so the verb must also be plural:

• piace = singular
• piacciono = plural

Compare:

• A mia sorella piace il broccolo. = My sister likes broccoli / likes the broccoli.
• A mia sorella piacciono i broccoli al forno. = My sister likes oven-baked broccoli.

The grammatical subject is i broccoli al forno.

That can feel strange to an English speaker, because in English we would usually make my sister the subject:

• My sister likes baked broccoli.

But with piacere, Italian treats the liked thing as the subject:

• i broccoli al forno = subject
• a mia sorella = indirect object

That is why the verb is plural.

Yes. That is completely correct.

Italian word order is flexible here. Both of these mean the same thing:

• A mia sorella piacciono i broccoli al forno.
• I broccoli al forno piacciono a mia sorella.

The difference is mostly about emphasis:

• A mia sorella... puts focus first on my sister
• I broccoli al forno... puts focus first on the oven-baked broccoli

Because with a singular close family member, Italian often omits the article before the possessive:

• mia sorella
• mio fratello
• mio padre
• mia madre

So after a, you get:

• a mia sorella
• not normally alla mia sorella

This omission is common with singular, close family terms when they are not modified.

Compare:

• mia sorella = my sister
• la mia amica = my friend

Family words behave a little differently from many other nouns.

Yes. In Italian, broccoli is plural.

• singular: il broccolo
• plural: i broccoli

So this sentence is literally talking about the broccolis / broccoli in a plural sense.

That is another reason the verb is plural:

• il broccolo piace
• i broccoli piacciono

Al forno means oven-baked, baked in the oven, or more literally in the oven style.

It is a very common Italian expression in food language.

Examples:

• pasta al forno = baked pasta
• patate al forno = roast potatoes / oven-baked potatoes
• broccoli al forno = oven-baked broccoli

Grammatically, al is the contraction of a + il.

Italian uses definite articles more often than English.

So where English says:

• My sister likes baked broccoli

Italian often says:

• A mia sorella piacciono i broccoli al forno

That does not necessarily mean a very specific plate of broccoli. It can also refer to broccoli in a general sense, depending on context.

This is normal Italian usage.

No, that is not correct.

That sentence tries to use piacere like the English verb to like, but Italian does not structure it that way.

Correct pattern:

• A qualcuno piace / piacciono qualcosa

So:

• A mia sorella piacciono i broccoli al forno.

Not:

• Mia sorella piace i broccoli al forno.

Be careful: Mia sorella piace by itself would mean People like my sister or My sister is pleasing/attractive, which is a very different idea.

Yes. You can replace it with le.

So:

• A mia sorella piacciono i broccoli al forno.
• Le piacciono i broccoli al forno.

Here le means to her.

This is very common in Italian, and often both are used together for emphasis:

• A mia sorella, le piacciono i broccoli al forno.

In everyday speech, that kind of doubling is very natural.

No, but it is the most natural and common way.

You could also use verbs like:

• adorare = to adore
• amare = to love
• preferire = to prefer

Examples:

• Mia sorella adora i broccoli al forno.
• Mia sorella ama i broccoli al forno.

But those are stronger or slightly different in tone.

If you simply want to say likes, piacere is usually the best choice.

It feels backwards only if you are thinking in English.

English:

• My sister likes broccoli.
  • subject = my sister
  • verb agrees with my sister

Italian with piacere:

• A mia sorella piacciono i broccoli al forno.
  • subject = i broccoli al forno
  • verb agrees with i broccoli al forno

So the verb is plural because broccoli are what is grammatically pleasing to your sister.

If the liked thing were singular, you would use piace.

For example:

• A mia sorella piace il broccolo.
• A mia sorella piace la pasta al forno.

If the liked thing is plural, you use piacciono:

• A mia sorella piacciono i broccoli al forno.
• A mia sorella piacciono le patate al forno.

So a good shortcut is:

• one thing → piace
• more than one thing → piacciono