You're talking about avidin, which is present in egg whites, and which destroys biotin in cats, not sure about dogs.
I wouldn't feed egg more than once in 2 weeks. But you may want to consider changing to wet food. It's really not much more expensive, because if you choose a nutritious brand (not the supermarket junk food), they will eat less. You can also supplement it with raw meat like I do and I save a lot of money that way. I'm actually considering going all raw, but I understand that's not acceptable to soem people.
There is really no need to feed dogs or cats dry food apart from our own convenience. I can only talk about cats, but I wouldn‘t really recommend feeding any dry food. Cats are designed to get their water from food. That's the way nature designed them, they have low thirst drive. When fed dry, they won't drink enough to compensate for the lack of moisture. They will only consume about 50% of the water they should be having. This can lead to kidney disease, UTI, crystals, blockage, renal failure and more. Especially if you have a tom, this is crucial. Male cats have a narrower urethra than female cats and are more prone to blockage from the crystals.
Free feeding also contributes to obesity. And the fact that dry food is over-processed means, that most of it’s little nutrition has been already destroyed, leaving almost no nutrients for your cat. It needs to eat more to meet it’s needs, and in the process consumes more calories from the fillers.
Btw wetting the dry food will not help. There’s bacteria on the kibble and the water would just allow it to grow.
The only way to give the cat it's natural hydration is to feed it wet food only.
But some wet foods are not of a very high quality, either. That goes for most commercial foods. Just like the dry, they are often made with cheap fillers such as corn, wheat, soy, rice etc. These are not a part of cat's natural diet (it’s an obligate carnivore – it eats meat) and they are not designed to digest it. Grain is carbohydrate which the cats can't process and it turns into blood sugar and fat, causing diabetes and obesity. In the wild, where cats only hunt for meat, diabetes and obesity are unheard of. It's us who cause these by feeding a species inappropriate food.
We usually read labels on our food, but rarely on the food for our cats. Learn to read the label and understand the ingredients. The healthiest food to feed apart from raw feeding is grain-free wet food with no by-product. Some good brands are Wellness CORE, EVO, Merrick, Nature's Variety, Blue Buffalo Wilderness and more. These will give your cat the proper hydration and nutrition it's designed to get and it will be strong and healthy.
If you switch it's diet, do it gradually, by mixing the current food with the new one over couple of weeks until there's only the new. This will prevent diarrhea and upset stomach.
You will probably get a lot of different answers, so google feline nutrition or look at the links below, and do the research for yourself. I personally wasn't able to find one reliable source (besides the pet food industry) that would say grain is beneficial for cats or that dry is beneficial for them.
More on cat nutrition below,