In a large saucepan, melt ¼ cup butter over low heat. Sprinkle with ¼ cup flour and whisk for 1 minute until combined.
Add milk, a little at a time, while whisking until the consistency is like a thick cream. You may only need 1 cup milk (up to 1.5 cups) depending how thick your bechamel looks. This should take a few minutes on low heat.
Add crumbled feta, shredded cheddar, parmesan cheese, lightly beaten eggs, salt, pepper, and dried mint. Keep stirring on low heat until cheese is melted and filling is combined (sauce will be slightly lumpy from feta which is OK). Remove from heat and set aside to cool.
Preheat oven to 350°F. In a microwave-safe bowl, melt ¼ cup butter for 30 seconds and then mix in ¼ cup olive oil. Using a pastry brush, coat the bottom and sides of your 9x13 baking dish with some of the butter/oil mixture.
Transfer thawed phyllo sheets to a flat work surface and cover with a slightly damp towel. Remove one phyllo sheet and place on the bottom of the baking dish, then brush all over with the butter mixture. Repeat with 6-7 more sheets of phyllo, arranging so the dish is covered (the sheets should overlap in the middle and spill over the sides of the baking dish).
Pour in the cheese mixture and spread out in an even layer. Fold the overhanging phyllo dough back over the filling and brush the top with the butter mixture. Layer 6-7 more phyllo sheets on top, brushing each with the butter mixture. Use your fingers to tuck the pastry edges in and under the pie, sealing in the sides.
Cut your cheese pie into 12 pieces before baking (this will make cutting and serving easier) and sprinkle the top with sesame seeds.
Place the baking dish in the preheated oven for 45-50 minutes or until the top is golden brown. Let cool for at least 10-15 minutes before serving so the cheese sets.