@Ryan_L makes a very good objection: there's little chance of this "double purpose jaw" evolving.
You can start with a human-type jaw. Then, we introduce an environmental change - something happens so that there's a big advantage in being able to swallow something whole. It can be anything, say a fruit. Somehow this fruit pops out for very little time and there's a race to who eats it first.
Being able to snatch it in a single mouthful becomes a marked advantage, but now the jaw gets stressed. Subluxation of the temporomandibular joint becomes frequent, and therefore, having an easily-fixed mandible is now advantageous.
Very soon in evolutionary terms, the mandible becomes only weakly attached to the cranium, but the masseter muscles develop into a sort of double trapeze: when you bite, first the jaw gets locked securely into place by the longer muscle, then leverage is applied by the shorter, more powerful transversal muscle.
Now you can take large bites, and still chew.
And it turns out that the fruit you eat doesn't get properly digested from the fruit's point of view (the "fruit" is a bribe the plant "pays" to animals so that they will defecate the seeds far and wide. It's fine tuned to attract, or be more visible to, the most useful animals). Seeds swallowed by our guys are less likely to sprout. Fruits partially eaten by other animals get a better chance. Which means that larger fruits have a selective advantage going for them: they don't get eaten whole, and/or get splattered every which way, seeds included.
There's now a selective advantage in having a wider mouth, or a more elastic mandibular joint: you can eat more fruit, of a kind your competitors aren't equipped to snatch away.
An "arms race" starts between the plants that grow larger fruit, and our guys who develop larger mouths.
After some time (waves hands quickly) the upper jaw also gets modified and can be partially extroflected, but now the oesophagus is stressed and the neck must start enlarging and becoming more flexible.
In the end, if everything goes exactly right, you might get something midway between a Great Shark mandible, a snake's jaw and a Xenomorph telescopic dentures. A second set of mobile jaws inside the first, like a snail's radula, can be used to "chew" the captured food so that it can be swallowed more easily.
The "eating" would then consist in our guy fully extending its jaws, until they are wider than its now toad-like head, and engulfing the food, remaining in a quite embarrassing and ridiculous position until the inner jaw "walks" the outer jaw to fully enclose the mouthful and start closing on the other side. Meanwhile, the inner jaw is also worrying at the food, scraping away matter that gets ingested. Finally, the mouthful is completely in, the underjaw fully distended, and our guy sports a large bulge under its chin and neck.
(This is the point where the jaw can become separated into two horizontal halves).
The inner jaw can maybe "rotate" slowly the mouthful to scrape all sides, so that it gets progressively smaller until it can fit in the distended oesophagus and finish its journey in the stomach.