Egg-Detection in Pastries: An Analysis in Heat-Alterated Molecule Identification
Egg white consists of mostly Ovalbumin. It is a protein, and so will be hard to distinguish from other proteins in a baked product. Egg white changes structure to a solid polymer on cooking. You could dissolve it in a mixture of detergent and hair perm, returning the Ovalbumin to its uncooked state, and then run it through a column to identify it.

There might also be biological assays you could run using enzymes/antibodies which would only interact with Ovalbumin.

I don't think there are any chemical means to identify a particular cooked protein easily.

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