An egg is 6.8 mg/gm of phe and this recipe does contain 15 gm of egg yolk (which is 8.19 mg/gm or 123 mg of phe). Keep in mind I am not recommending that your child have an egg for breakfast. The egg in this recipe is making 82 cookies so the protein is distributed throughout the recipe (adding only 1.5 mg phe per cookie). Can you work the cookie into your child's phe allowance? (If not, then there is a tip at the bottom of the recipe that tells you how to lower the phe content.) I promise as long as your child does not have an egg allergy and if you can work it into his allowance, it is safe.
You will find that Cook for Love recipes sometimes contain a small amount of flour in baking, and almost always contain butter and heavy cream (other ingredients considered forbidden by many), as opposed to non-dairy creamers and margarine that are lower in phe. Why do I use eggs and other foods that are considered by many to be bad/off limits for the low protein diet? I use them if they make a significant difference in the taste/texture, if the end product is still a reasonable amount of phe, and if it means I am giving Molly (my daughter) less chemicals.
You are not making a specialcookie for just your child, my goal here is that that everyone in your family enjoys this recipe. You do not need to make a high and low protein version. No one will know that this recipe is made for a special diet, they will just know it tastes great. Cook for Love is about starting new family traditions that your child can be a part of, that you share together, the way it should have been.