In a large, heavy skillet, heat the 3 tablespoons of olive oil and cook the onions and garlic. Add the chicken and cook over medium heat for about 7 minutes, until the chicken browns lightly.
Meanwhile, finely chop the herbs.
Remove the skillet from the heat and mix in the herbs. Season with salt and pepper. Toss the mixture with ¼ cup each of the graviera and kefalograviera. Set aside until ready to use.
Lightly oil a 14 -or 15-inch X 2-inch deep round baking or pizza pan, and preheat the oven to 350 F (180C).
Layer 5 phyllo sheets, one at a time, in the pan. Try to get the sheets even to the edge and brush each sheet with 1 tablespoon of olive oil.
Spread the chicken filling evenly over the phyllo. Place three sheets of phyllo over the chicken, brushing each with olive oil and spreading them so that they are fanned and hanging over the edge of the pan as evenly as possible.
Brush the overhang with a little olive oil. Bring together the top and bottom overhanging phyllo and roll it together to form a ring or rim inside the pan around the circumference of the pie.
Break the eggs into a medium size mixing bowl. Whisk the semolina slowly into the eggs and keep whisking until the mixture is smooth and without any lumps. Whisk in the chicken broth and remaining cheeses. Season if desired with a little salt and pepper.
Score the pie into serving-size pieces with a sharp knife. Pour the egg-semolina mixture evenly over the surface of the pie and tilt the pan so that it goes all over. Place the pie in the preheated oven and bake until the phyllo is golden and the top set and nicely browned for 45 to 50 minutes. Remove from the oven, let cool slightly, and serve.