White pasty egg cover Help PLEASE!

Yesterday when my wife collected the eggs from the coop, the one egg from our best layer was covered with white pasty stuff. I thought it was from our rooster since he likes to try to mate with the hens while they are laying, but I want to make sure so I turned to BYC for help. What could it be and should I be worried. Her vent is clean and normal looking when I checked it and she is eating and drinking fine. She is also pooping normal also and acting normal. What is it? I appreciate all help on the matter.

Some good clear well lit pics would help.
Hmm.... have never heard of a cock/erel mating while a hen/pullet was laying.
ROLF, I guess it's possible.

Not sure what you mean by 'pasty'...like sticky/wet/tacky?
Could be urea(the white part of chicken poop).
Could be excessive bloom......or membrane.

Need pics.

She retrieved the egg so I have no idea what it looked like but she said it was tacky but washed off easily and was white. I have picked up eggs that just fell out of the vents before with the clear liquid on them and then left a finger print on the brown egg as a result. I was just making sure she wasn't sick. She is acting fine and did just take 2 days off from laying which I contributed to winter. She gave 1 bloody egg about 2 months ago after 3 days of not laying and I found where someone said that it was normal so I wasn't worried after it went back to normal.

Most of the time my hens will lay clean eggs, but sometimes very small traces of dung is on them, but never large amounts. My rooster has a notion to sleep in the nesting box and has, believe it or not, set on eggs overnight if they were laid very late. Last week he sat on an egg and split it apart stepping on it, but I cleaned it up fast so they didn't get a taste of the insides.

I feed my chicken the dumor scratch feed from tractor supply along with cracked corn and wild bird seed from walmart. The egg shells are very solid all the time and the yolks are very dark compared to the store eggs. I didn't think it was in their feed habits, but you never know sometimes.

Talking about my rooster. My rooster is very not that bright to say the least. He is unable to grasp the concept of many things since we got him in August at the age of 14 weeks. He was (and still does sometimes) hides from the hens. They bully him around a lot and he lets it go, but does stand up for them when needed. He never had a perch to stay on at night until we got him so he can't balance when I put him on the pole. He can't stand and mate well on the hens' back. I think he may be too big for the hens so I am getting 2 more RIR hens that are 1 year old to possibly give him a more compatible mate.
He finally learned to crow this morning, it went ooh...eee...ooh. Sounded like a monkey to me, I thought one of my kids were making weird noises until I figured out it was him.