If undercooked eggs are a concern, you’ll want to make sure to heat the oat bran up to 165 degrees to eliminate any risk of salmonella with the eggs.
For a protein boosting alternative, I sometimes add collagen protein powder to this as well. (affiliate link)
The longer you let the oat bran simmer, the more it will thicken and absorb moisture. I usually add more water or milk after adding the eggs, just before serving, to make sure it stays creamy and doesn’t get lumpy. The texture reminds me of Cream of Wheat!