Begin by placing a large pot on the stovetop. Combine thinly sliced onions, butter, and a drizzle of olive oil, sautéing over medium heat until they become translucent. Gently add a whole chicken, ensuring it's covered with water. Bring the mixture to a boil, then decrease the heat, allowing it to simmer for approximately 1.5 hours until the chicken is fully cooked and tender.

Once the chicken has poached, carefully remove it from the pot and let it cool slightly. Strain the liquid, reserving the onions and discarding any solids. Reduce the poaching liquid in the pot until it reaches a gelatinous consistency. Shred the chicken meat once it's cool enough to handle, then mix it with the caramelized onions, fresh eggs, and a pinch of black pepper in a large bowl, combining thoroughly.

Preheat your oven to 230°C (446°F) and grease a pie dish generously with olive oil. Begin layering the filo pastry by placing three sheets in the dish, brushing each one lightly with olive oil to keep them moist. Spread the chicken filling evenly over the pastry. Then, layer three more sheets atop the filling, again brushing with olive oil.

To finish the pie, seal the edges by folding the pastry over itself, then brush the top generously with olive oil. Place the assembled pie in the preheated oven and bake for 15 minutes until beautifully golden. Lower the oven temperature to 200°C (392°F) and continue baking for an additional 45 minutes.

Once baking is complete, remove the pie from the oven and allow it to rest for about 10 to 20 minutes. This resting period enhances flavors and makes slicing easier. Cut the pie into wedges, serve warm, and relish the delightful layers.