Heat oil in a pan and braise the chopped onions and carrot. Add the chopped chicken meat with a little water, so it doesn't stick and saute briefly.
Prepare the sauce. In a suitable container, heat the oil and butter for it. Pour in the flour and stir so it braises slightly, pour in the milk and water. Stir until slightly thickened, season and it's ready.
In a bowl, place 2 phyllo pastry sheets, leaving the edges to dangle freely on the outside. Fill with part of the stuffing and bring the edges together on top. Turn it over onto an oiled tray - you're going to get a ball out of it.
Repeat this process another 3 times - for a total of 4 servings. Pour the sauce over the balls and sprinkle generously with grated cheese. Bake in a preheated 392°F (200 °C) oven until golden.