Use a food processor to finely chop the frozen, grated filo dough. The strands should be about the size of a grain of rice. Pour the dough into a large mixing bowl. In a separate bowl, mix together the ricotta, mozzarella and 1/3 cup sugar.
Place the butter in a large liquid measuring cup or bowl with a spout. Heat the butter in a microwave until completely melted and let it sit for several minutes until a thick white foam has formed on top. Clarify the butter by using a spoon to skim off the foam.
Carefully pour the butter into the bowl of filo dough. Avoid pouring in the white milk solids at the bottom of the clarified butter. Use your hands to mix the butter and dough together. Make sure that the butter is absorbed by taking handfuls of the dough and rubbing it between your palms.
Evenly spread the buttered filo dough into a 20x30cm pan and firmly press it into the bottom and edges. Spread the cheese mixture onto the dough, avoiding the edges of the pan.
Bake in the preheated oven until the cheese is slightly golden and the edges of dough are brown and bubbly, 30 to 35 minutes.
While the filo is baking, prepare the syrup. Combine the water and 1/2 cup sugar in a small saucepan. Bring the mixture to a boil over medium-high heat. Reduce the heat to medium and stir in the lemon juice. Simmer, stirring constantly, until the sugar is dissolved and the mixture is thickened, 5 to 7 minutes. (Do not let the mixture turn golden and caramelise.) Remove from heat and add the rose water; set aside.
Remove the filo from the oven. Place a large platter or baking tray over the baking dish. Using oven mitts carefully invert the baking dish onto the platter so the filo is on top. Pour the syrup over the filo. Cut into pieces and serve while hot.