I remember that scene as well. IIRC, it wasn’t that phasers wouldn’t work, but that it would set off some kind of security alarm if someone fired one off on board.
That doesn’t explain any phaser-based coffee-warming not setting off the alarm, but maybe Enterprise’s internal security system was upgraded at some point between “The Corbomite Maneuver” and Star Trek VI.
Here’s the scene in question.
Now I’m wondering why there are phasers stored in the ship’s galley.
Jesus, how I hate those marching band uniforms. Yeccch! 
No one can fire an unauthorized phaser aboard a Starfleet vessel, but there just happens to be a chest full of them on the wall of the galley? Niiiiiiiice!
Everybody knows capturing the galley has top priority for any enemy boarding party, right after the Bridge, Engineering, and Auxiliary Control. 
I’ll bet that’s no galley. It’s way too small to feed the entire ship. It might be a “hobby room” much like an art studio or a metal shop. Phasers would be there for immediate access rather than a central arms room.
I scramble/beat my eggs, pour them into pan, then add cheese, etc. Once mostly done, I flip half over. So, there is scrambling done, but not once in pan.
IIRC, Robert Picardo (hologram doc from Voyager) said that the most important quality for casting Star Trek actors is that their butt looks good in spandex.
“Kitchenette,” then. Even more important for a boarding party to capture the ship’s kitchenette! 
I thought he was the Walrus.
Starfleet learned some things from the early missions, such as that maybe you want to know if weapons are being fired inside the pressure hull, just so you can seal off anyone trying to take over.
(And I suppose it just happens that the quick-access phaser storage for that section of that deck is in the auxiliary galley; in another it may be in a conference room and in another it may be in the heads. I do hope they are also smart-chipped to only fire if someone authorized is holding them.)
BTW in TOS they did show a few times that the phasers had more settings between “stun” and “completely vaporize”. But I never saw the one for “vaporize only the metal outside, leave contents intact”. Those are some fine tuned weapons!
That would be the “script” setting.
Also, considering that depending on your timeline there have been a few global catastrophes along the way, it is conceivable the knowledge of the omelet as a distinct thing has been lost, as was the knowledge of surge breakers and seatbelts.
Star Trek technology design is based on the principle of WYWIWYG - What You Want Is What You Get.
An early indication of just how batshit crazy Lt. Valeris really was.
If they let Reg Barclay roam around the ship unattended, anything is possible.
My link didn’t work, but there are obvious visual differences between an omelet and scrambled eggs. How people in the 21st century confuse the two is beyond me.
In case the pilot lights for the burners go out?
Can I ask a side question?
Ok, they gave us six epis of Discovery. In the third the protagonist is being blamed for causing the deaths of 8000 Federation crews. Huh? How? Her clever plan to do a Vulcan didnt work. How did the protagonist cause the war?
And boy the stupid. They beam on board the Klingon ship- with a whole team of security, armed with the high powered phaser rifles? Nope, just the two of them, with pistols.
Then later, they board that derelicts- again, pistols, not rifles. But they have them, they were seen.
This makes me wonder what happens if they use this on a person. Would it only vaporize their skin, leaving their innards to just flop to the floor? That reminds me of the transporter in Galaxy Quest! Seems like a lot of potential cleanup for the galley so I’m surprised the head chef doesn’t have a “If you use it, you mop it!” sign on the box. On the other hand, maybe vaporizing the pots and pans is how they manage dirty dishes. Does the Enterprise employ dish washers?
I looked up the original scene in question on Youtube and the title is “Commander Riker makes Scrambled Eggs”. And you’re right, these are very clearly scrambled eggs and despite his claim to being an artist, they’re dry and overcooked. Yuck! Granted, they’re alien eggs so maybe they were perfectly prepared for that style.
