Learn Turkish: Ben pilavı tencerede pişiririm.

The -ı is the specific direct object ending (often called the accusative case). Because the sentence means "I cook the pilaf" (a specific batch of pilaf) rather than just "I cook pilaf" in general, we add -ı to pilav.

What is the difference between pişiririm and pişiriyorum?

Pişiriyorum is the present continuous tense ("I am cooking right now"), while pişiririm uses the simple present tense (the aorist). The -ir ending tells us this is a habit or a general statement, meaning "I (generally) cook the pilaf in a pot."

The dictionary says "to cook" is pişmek, so why does this verb have an extra -ir- before the tense ending?

In Turkish, pişmek means "to cook" by itself (for example, "the food is cooking"). When you are cooking the food, you add -ir to the root to make it pişirmek (literally "to cause to cook"). So, the first -ir makes it an action you do to the food, the second -ir is the simple present tense, and -im means "I".

Why is the subject Ben included if the verb already ends in -im ("I")?