Not sure if I understand your node setup. The first Mix Shader node is set to 0, so it exclusively uses the top input and the Glossy shader is basically useless. And then you mix the Glass shader with a Transparent shader based on a constant mixing factor?
Another thing is: “Always give reflective surfaces something to reflect.”
Your egg white is a planar surface in what seems a to be a plain grey environment. This can only result in plain grey (= mostly invisible) reflections.
Here’s quick and dirty try:
It occurs to me that the egg white is not of uniform color. It seems to be more yellowish-orangish around the yolk until it is almost watery-clear (with only a hint of yellow) at the borders. Therefore I painted an area around the yolk black in vertex paint mode with a large brush.
This vertex paint can be accessed in Cycles via a Attribute > Col node. This node is used to drive:
a) The color transition within the Glass shader and
b) the mix between the Glass and a Translucent shader, to also have the egg white slightly more opaque around the yolk:
Perhaps this can serve as a starting point?
BTW, egg shells model taken from this blend.