The riddle
What Is a Turkey's Favorite Dessert?
What a cobbler is
A cobbler is a deep-dish fruit dessert with a thick crust on top (Merriam-Webster). Peach is one of the classic fillings, so peach cobbler is a familiar name. The same word also names someone who mends shoes, which is its older use. Dictionary.com describes the crust as a rich biscuit crust, usually only on top, which sets a cobbler apart from a pie with a crust above and below. The riddle changes one sound in the dessert and gets a different word.
What a gobbler is
To gobble, in one dictionary sense, is to make the sound of a male turkey, and a gobbler is a male turkey (Merriam-Webster). The same verb also means to eat greedily, which suits a hungry bird. Merriam-Webster records the turkey-sound sense from 1680.
One sound apart
Cobbler begins with a k sound and gobbler with a g sound. Linguists describe the two as a voiceless and a voiced pair made in the same place at the back of the mouth. Put a hand on the throat: g vibrates and k does not. Everything after the first sound is identical, so cobbler and gobbler rhyme. The peach stays the same, and the whole joke rests on that single swap.
Inventing more
Any other dessert that can shift by one sound could belong to another animal. A group might try one for a cat or a goat and see how far the sound trick stretches. Most attempts fall apart because the new word has to be a real word for the animal as well.