Learn Farsi: چشم‌هایت زیبا است.

It is the -at ending. To say 'your', Farsi often attaches the possessive ending -at to the noun. Because cheshm-ha (eyes) ends in the vowel 'a', we add a 'y' sound as a bridge to make it easier to pronounce, resulting in cheshm-ha-yat.

Since 'eyes' is plural, why does the sentence use the singular verb ast (is)?

In Farsi, when a plural subject is not a human being (such as objects, concepts, or body parts), it almost always takes a singular verb. Since eyes are body parts, the verb stays singular (ast) rather than changing to the plural hastand (are).

Why isn't there an ezafe connecting the noun to the adjective ziba?

You only use an ezafe (the 'e' or 'ye' connecting sound) when an adjective is directly describing a noun inside a phrase, like 'your beautiful eyes'. In this sentence, ziba is stating a fact about the noun using the verb 'to be'. Just like in English, you don't need a connector for 'Your eyes are beautiful'.

How would this sentence sound in everyday, casual spoken Farsi?

In casual speech, the possessive ending -at usually changes to an -et sound, making it cheshm-ha-yet. Furthermore, ziba ast naturally blends together into zibast. So in conversation, a native speaker would usually say: cheshm-ha-yet zibast.