In Roumeli, in the center of Greece's mainland, yiouzlemedes are savory cheese-filled pastries. In Lesvos, on the other hand, the pastries are sweet and deliciously excessive. They are stuffed with cheese and mastic and sometimes sprinkled with still more cheese and ground mastic.
Combine the water and sugar in a medium-sized saucepan and heat over medium heat. As soon as the syrup begins to boil, reduce the heat to medium-low, add the lemon juice, and simmer for 10 minutes. Remove from the heat, set aside, and let cool.
Lightly dust a work surface with flour.
In a medium-sized bowl, combine 2 cups of the flour and the salt. Make a well in the center of the flour and pour the water and olive oil into the well. Little by little mix the flour into the liquid. Knead by hand, either on the work surface or in the bowl, until a stiff, smooth dough forms. This will take about 10 minutes. Add a little more flour or water if necessary during kneading if the dough seems either too loose and sticky or too stiff. (The pastry may also be made in the bowl of an electric mixer outfitted with a dough hook.)
Combine the egg and cheese in a small bowl and mix in the cinnamon. In a mortar with a pestle, pound the mastic with the sugar. Stir into the filling.
Divide the pastry dough into 2 balls. Lightly flour a work surface. Pat down the first ball and roll it out with a lightly floured rolling pin to a circle about 20 inches in diameter. Using a glass or round cookie cutter about 3 inches in diameter, cut circles out of the dough. Place 1 teaspoon of the filling on each circle, then fold over to form a half-moon. Pat down the edges with your fingertips or more decoratively with a fork to seal. Collect the remaining dough, knead it, and roll it out again. Continue cutting out circles and filling them until the dough and filling are used up.
Heat about 2 inches of oil in a large, heavy frying pan over medium-high heat. When the oil is very hot, fry the yiouzlemedes, a few at a time, in the hot oil, turning once, until golden on both sides. Remove with a slotted spoon and let drain on paper towels. Place on a large serving platter and drizzle generously with the sugar syrup. Dot with the crumbled cheese and sprinkle with cinnamon. Serve immediately.