Learn Italian: Dopo aver cenato, quello che gradisco è un dolce. - After having dinner, what I appreciate is a dessert.

In Italian, you cannot use cosa (which means "what" in questions) to mean "that which" or "the thing that" in a statement. To emphasize a specific thing using a pseudo-cleft structure (What I appreciate is...), you must use quello che (literally "that which") followed by the verb, and then the verb essere to introduce the focus of the sentence.

Could I use ciò che instead of quello che here?

Yes, ciò che and quello che are perfectly interchangeable in this structure. Both mean "that which" or "what" in a declarative sentence. Quello che is slightly more common in everyday spoken Italian, while ciò che can sound a tiny bit more formal, but both are completely natural and correct here.

Why does the sentence use Dopo aver cenato instead of a conjugated verb like Dopo che ceno?

When the subject of the action is the same in both parts of the sentence (you are the one dining, and you are the one appreciating dessert), Italian typically follows dopo with a past infinitive (avere or essere + past participle). Dopo aver cenato literally translates to "After having dined". Also, notice how avere drops its final "e" to become aver; this is an optional but very common habit to make the phrase flow more smoothly.

I usually use for "I like". How is different here?