The riddle
What Do You Call a Running Turkey?
The word fast
The word "fast" has two meanings in Merriam-Webster that point in opposite directions. One is quick, as in a fast runner. The other is firmly fixed, as in stuck fast or fast asleep. The riddle uses the first: a turkey on the run is a fast turkey. Then the phrase "fast food" arrives, and a description turns into a name. The two senses are less far apart than they look. The Online Etymology Dictionary says the older meaning is firm or fixed, and the sense of quick grew out of using the word for doing something firmly and hard, as in run fast.
Reading fast food
Merriam-Webster defines fast food as food prepared and served quickly, and as the restaurants that do it. The riddle takes the two words apart. "Fast" describes the turkey's speed, and "food" supplies the noun the phrase needs. Put them back together and a bird in a hurry becomes a menu term.
Running has many jobs too
"Running" can mean moving quickly on foot, but it also describes flowing water, a working engine or a person running for office. The question works because the first meaning is the obvious one, and the listener has to follow it all the way to the end of the sentence before the second word of the answer clicks.
Telling it with timing
Stress "running" slightly and then pause. A listener who guesses a funny name for a bird will go through a few tries, and the answer is easier to enjoy when the last word is delayed by a beat. The joke needs no detail about the turkey itself, because it belongs entirely to the words.