Learn Finnish: Minun tekee mieli jäätelöä.

This is the core of the mielitekorakenne (structure of desire) being taught in this lesson. In Finnish, you don't directly "do" the craving. The literal translation is closer to "My mind makes for ice cream." The word mieli (mind) is the actual grammatical subject, which is why "I" becomes the genitive owner of the mind (minun).

Why is jäätelöä in the partitive case here?

When you use the tekee mieli structure with a noun, the object of your desire must always be in the partitive case. While ice cream is often an uncountable mass noun anyway, this rule applies to countable things too. For example, craving a new car would be Minun tekee mieli uutta autoa. The craving is treated as an abstract, unbounded feeling, which triggers the partitive.

If I want to say "We crave ice cream," does the verb tekee change to match "we"?

No, the verb always stays in the third-person singular form (tekee) regardless of who is experiencing the craving. Because the grammatical subject is always the singular word mieli, the person craving just changes in the genitive slot at the beginning: Meidän tekee mieli jäätelöä. The verb only changes if you change the tense, such as the past tense (teki mieli).

Can I leave out the word minun like I can leave out minä in standard sentences?