Is my hen sick?

My sister and I just watched our Fayoumi Ace suddenly go from normal perky roadrunner to fluffed up lethargic hen. She was walking around the chicken pen, straining, occasionally dripping what looked like egg whites, then stopped and laid her egg in the middle of the yard. It took a few minutes and we had to shove away curious hens while she kept pushing, sometimes making noises of I think pain.
Once it came out... well, this is what she laid:



After a little bit, she walked off and went back to being herself.

We removed the membrane/stuff around the egg, and it was a completely normal, fully formed egg with a beautiful orange yolk.

I'm thinking she was almost laying an egg within an egg, but the solid-ish white stuff looks weird to me. Anyone know what it is?

 

Layer pellets, plus occasional kitchen scraps. And, well, whatever she finds when I let her free-range.
Ace is one of several hens I have who like eating gypsy moth caterpillars. I mean, she doesn't eat that many, but not all hens are interested in those.

 

Thanks for the link! It didn't quite help me figure out what the lumpy-looking white stuff was, though. I'm still not sure whether it's lumps of calcium or a weird infection or what.

 

She's just turned 1 year old. She's been laying since... maybe November or December? I have a total of 3 Fayoumis and they've been really consistent with the eggs they lay until now. Just regular little white eggs. No outliers in size, shape, nor texture.

Her behavior today is normal, and I will examine her vent tonight once it's dark.

We just had a heat wave last week. If that can cause hens to stop laying, can it cause "glitches" like this?