Aspiration Pneumonia: This is the most likely diagnosis given the symptoms of aspiration of eggs and copious sputum production. Aspiration pneumonia occurs when foreign material is inhaled into the lungs, which can lead to an infection. The presence of eggs suggests that the aspiration event involved a solid or semi-solid substance, which is consistent with aspiration pneumonia.

Chronic Bronchitis: This condition is characterized by chronic inflammation of the bronchial tubes, often resulting in copious sputum production. While it doesn't directly explain the aspiration of eggs, it could be a contributing factor to the overall respiratory condition.
Bronchiectasis: A condition where the airways of the lungs become abnormally widened, leading to a buildup of excess mucus, which can result in copious sputum production. Aspiration could be a complication or a contributing factor to the development of bronchiectasis.

Foreign Body Aspiration: Although the question mentions aspiration of eggs, it's crucial not to miss the possibility of other foreign bodies being aspirated, which could lead to airway obstruction or other serious complications.
Pulmonary Abscess: A potential complication of aspiration pneumonia, where an abscess forms in the lung tissue. This condition requires prompt diagnosis and treatment to prevent severe outcomes.

Cystic Fibrosis: A genetic disorder that leads to severe respiratory and digestive problems. While it's rare and not directly linked to the aspiration of eggs, it could be considered in the differential diagnosis for copious sputum production, especially in younger patients or those with a family history of the disease.
Lipoid Pneumonia: A rare condition caused by the inhalation of lipids or oils into the lungs. If the eggs aspirated contained significant amounts of lipid material, this could be a rare but possible diagnosis.