Before you start making the pie, let the phyllo within its package at room temperature for about 1 hour.
Preheat oven at 220°C / 428°F.In a mixing bowl, add the rice flour. Start pouring the milk while stirring constantly with a hand whisk until a thick mixture forms. Stir in the Feta and Anthotyro cheese as well.
Mix together the melted butter with the corn or sunflower oil. Keep aside along with a pastry brush.
Take a half sheet pan (18 x 13-inch / 46 x 33 cm) or your oven's tray pan and place it on your working surface. Grease the pan well with the butter and oil mixture.
Unwrap the phyllo from its package and lay it open next to the pan.
Take two phyllo sheets and place them in the pan. The phyllo should cover the height of the pan on one side. Grease well with the butter mixture again.
Then add two more phyllo sheets covering the height of the pan on the opposite side this time. Grease again with butter mixture.
Now add one phyllo sheet covering the center of the pan crossing the other ones and covering one of the uncovered edges of the pan. Grease with butter and add one more phyllo exactly the same way covering the height of the last edge of the pan that was left uncovered.
Grease with butter and add the filling on top. Spread it evenly with a spoon.
Place all of the remaining phyllo sheets two by two (no need to cover the edges this time). Greasing well with the butter mixture after every two phyllo sheets. It doesn't matter if they tear or crumble a bit.
Once you have placed all of the phyllo sheets, overlap and tuck in the edges, to create a pie ending on all sides. Sprinkle all of the remaining butter on top. Add 1,5 tablespoon of water in the bowl or pot you kept the melted butter and scrape any of its leftovers, with the pastry brush. Then drizzle this water all over the pie. With the pastry brush gently tap on top for the phyllo to stick together.
Thawed the top phyllo's with a knife, dividing into 16 square pieces (don't let the knife reach all the way to the bottom). Bake for about 20 minutes until golden brown on top. Shake the pan a bit to see if the pie moves in the pan, to make sure it's baked at the bottom as well.
Cut and serve the pie. Once the pie has reached room temperature you may keep it inside your oven or in a food container for up to a day at room temperature (20-25 degrees Celcius). It won't spoil because the feta cheese is salty enough to preserve it. If you have leftovers beyond that, that you need to refrigerate, then you may reheat the pie in the microwave on medium after taking it out of the fridge.