Learn Turkish: Hamile kadının aniden baklava ve turşu yiyeceği tuttu. - The pregnant woman suddenly had an urge to eat baklava and pickles.

The "sudden urge" construction (-(y)AcAğI tutmak) works like a possessive compound. The action she has an urge to do is treated as a possessed noun ("her future eating"). Therefore, the subject takes the genitive case (kadının) and the verbal noun takes the possessive suffix (yiyeceği). A literal translation would be "The pregnant woman's going-to-eat took hold."

How exactly is yiyeceği formed from the verb yemek here?

It combines the root ye-, the buffer consonant -y-, the future participle -ecek (which softens to -eceğ-), and the 3rd-person possessive -i. Importantly, the root vowel e in yemek narrows to i because it comes right before the buffer -y-. So, ye + y + ecek + i becomes yiyeceği (rather than yeyeceği).

Why don't baklava and turşu take the accusative case, like baklavayı ve turşuyu?

Because the woman is craving baklava and pickles in a general sense, not specific ones that were previously mentioned. They act as indefinite direct objects of the verbal noun yiyeceği, so they remain in their bare dictionary form and are placed directly before the verbal noun.