Prepare the eggplant. Cut off the tops and bottoms. Then dice into small pieces. Soak in plenty of water for at least 30 minutes. Then drain in a strainer.
Make the filo dough. In a bowl, whisk together the flour, salt, and baking powder. Pour in the olive oil, then slowly add the water while kneading the dough. It should feel sticky but without leaving fragments on your hands.
Rest the dough. Shape the dough into a ball. Dust the top and bottom with flour. Cover the bowl with a dampened tea towel or plastic wrap and rest at room temperature for 30 minutes.
Heat a generous amount of olive oil in a medium-sized cooking pot or large saute pan.
Saute the onions then add the eggplant in batches. Season with salt. Cook until soft and mellow. The more you cook the eggplant at this point the stronger its flavor will be inside the filling.
Add the red peppers and cook for a minute.
Stir in the tomato paste and crushed tomatoes. Cook until the filling drains to the point where it starts to get sticky at the bottom. Take the pot off the heat.
Add the dried oregano, paprika, and basil. Stir and set aside to cool.
Preheat the oven to 180°C / 350°F.
Shape the filo dough into a log and cut it into 6 pieces. Each piece is a filo. 2 for the bottom, one between the filling layers, and 3 for the top of the pie.
Shape each piece into a round ball and keep aside, covered.
Make the first filo. Dust the work surface with flour and place the dough ball. Dust the dough ball with flour also. Start rolling in all directions with a rolling pin until you have a round filo big enough to cover the bottom and sides of the pan. Always dust the bottom and top of the filo with flour as you roll, so it won't stick. Also, flip the filo as seen in the video below.
Brush a 31 cm (12-inch) round pan with olive oil.
Roll the filo around the rolling pin and then unroll it inside the pan. Gently pull it a bit to cover the sides. Brush with olive oil.
Make the second filo. Make another filo, this time the size of the pan's bottom. Add it to the pan and brush with olive oil.
Stir the Feta and Kefalotyri cheese into the filling.
Spread half the filling in the pan.
Top with another filo. Brush it with olive oil.
Add the remaining filling.
Top with 3 more filos. Brush each one with olive oil. Overlap the edges of the bottom filo to create a nice pie edging.
Cut the pie into 16 pieces (try not to cut the bottom filo) and sprinkle with sesame seeds.
Bake for about an hour on the lower rack until golden brown and crispy.
Rest the pie at room temperature for at least 30 minutes before cutting and serving.