What changes the food answer
If your bird's species and health make egg appropriate, serve a very small plain portion and remove leftovers quickly.
Keep it plain
Use fully cooked egg with no salt, pepper, butter, oil, milk, cheese, onion, garlic, seasoning, or table scraps mixed in.
Portion by bird size
A budgie-sized portion is tiny. Large parrots can still overdo rich foods if egg becomes routine.
Watch hormones and spoilage
Warm soft foods and rich foods can encourage breeding behavior in some birds, and egg can spoil fast in a cage.
Better use
Treat cooked egg as an occasional tiny extra when it fits the bird, not as a daily protein solution.