In a medium frying pan heat the olive oil and sauté the onion in it for 2-3 minutes. Add the minced meat and stir continuously, separating with the wooden spoon. You could also add some water for easier separation of the meat. Cook until there are no raw pieces in the meat and majority of the water is evaporated (about 20-30 minutes). Add salt and pepper and remove from the heat. Let cool for 20 minutes.
Add the eggs and the grated cheese to the cooked meat and stir until well combined.
Preheat oven to 200 degrees C. Line a large baking tin with a sheet of parchment paper.
Extend the filo sheets on a flat surface. Brush the top sheet with the melted butter. Place about 2 tbs of the prepared filling on one side of the buttered sheet and smear it all around to cover about half of it. Try to avoid putting too much liquid from the filling onto the sheet - let majority of the liquid remain in the pan as it will be used at the end.
Take together the top filo sheet and the one underneath and roll them up, starting from the side with the filling. Roll the 2 sheets together into a log and then roll the log around itself like a snail shell. Place the roll into the prepared pan and brush it with butter again.
Continue the process until using all the filling
and all the filo sheets.
At the end you should have some liquid remaining on the bottom of the pan. Combine it with the remaining melted butter and brush once again generously all the rolls with the mixture. They should be well soaked when placed in the oven.
Bake the rolls for about 20-30 minutes or until golden brown.
In case the remaining liquid in the pan or the remaining melted butter are not enough to moist well the rolls, do not hesitate to add some milk or more melted butter.