Combine the sugar, water, lemon juice, and cinnamon stick in an 18 cm / 2-quart saucepan. Bring to a boil, then lower the heat and simmer for 5 minutes, until slightly thickened. Set aside to cool while you make the filling and dough.

Melt the butter in a 20 cm / 8-inch saucepan over medium heat. Add the semolina and toast for 2 to 3 minutes, stirring often, until it smells nutty and turns very light gold.

Add the sugar, water, and cinnamon. Cook, stirring constantly, for 4 to 5 minutes, until the mixture thickens into a paste that pulls away from the sides of the pan. Spread it on a plate to cool completely, about 15 minutes.

In a mixing bowl, whisk together the semolina, flour, sugar, salt, and baking powder. Add the oil and mix with your hands until crumbly.

Pour in the warm water and knead for 4 to 5 minutes, until you have a smooth, soft dough that isn't sticky. Cover the bowl and let it rest for 20 minutes.

Divide the dough into 12 pieces, about 40 g each. Roll each piece into an 8 cm round, about 3 mm thick.

Place a heaped teaspoon of the cooled filling in the center of each round. Fold in half to form a half-moon and press the edges together, then crimp with a fork to seal tightly. Wet your fingers slightly if the dough won't stick.

Heat the 750 ml of oil in a 24 cm / 9.5-inch deep skillet or pot to 170 C / 340 F. Fry the pastries in batches of 4 to 5 so they don't crowd the oil, turning once, for 3 to 4 minutes per side, until deep golden and crisp.

Lift the pastries out with a slotted spoon and let the oil drip off for a few seconds. Dip each warm pastry into the cooled syrup for 20 to 30 seconds per side, then set on a wire rack.

Arrange the shamishi on a plate and sprinkle with ground cinnamon or sesame seeds. Serve warm or at room temperature.