Glad I've been a help. That's what this site is for.
You don't *have* to have the cayenne (powder or crushed, either works) in the scrambled eggs, it's just one of the things you can give them to help perk them up. Just pick some up next time you get a chance. It's great just to have around, in fact. Just like garlic (again, powdered or fresh). Oh, and cinnamon powder, too. And, let's see... ah, apple cider vinegar - unpasteurized, with the "mother" in it! That's a natural antibiotic, as well as great to help them get their tummies (well, okay, crops and gizzards... digestive system) into a better Ph level to fight off germs and such. I keep a separate container of ACV for my critters than what I use for cooking or cleaning. And it needs its lid opened so it has fresh air every so often, too, to keep the mother in it healthy. All of these things - the ACV, the honey, the garlic, the cayenne, all are part of my critter first aid care, along with actual medications.
And as for things to help get everyone through molt/moult (no, Aus/Br English doesn't bug me... too many years having a huge addiction to shows like Doctor Who for a little difference in spelling or pronunciation bother me too much, LOL!)... I find my chooks LOVE cat food... dry food or moist (bagged/canned) or treats. It's extra high in protein and vitamins, which is exactly what they need for their bodies to get all those new feathers made! Get a little 5 pound bag of dry feed or a case of the canned feed, and pass it out once a day for a few weeks. Yes the canned food gets messy, but they'll be thankful. In fact, that might be something you can get Little (is that her name? hope I got it correct) interested in eating!
Oh, a few other goodies you might be able to treat her with - yogurt (plain, as it has no artificial sweeteners or other crud in it), and oatmeal! Either can also have fruit and honey added to it, in fact, for a bigger boost. And the honey needs - if at all possible! - to be raw, unfiltered, unpasteurized, just like it is when it comes out of the honeycombs... that's the kind that has the most antibiotic properties, the highest natural energy content, and the least extra, unnatural added crud in it.