Combine 1 cup sugar, 1/2 cup water, and the lemon juice in a small saucepan. Bring to a gentle simmer, stirring just until the sugar dissolves. Simmer for 5 to 7 minutes, then remove from the heat and let the syrup cool while you make the cake.

Preheat the oven to 350°F. Lightly grease a 9x13-inch baking dish. Working with 2 sheets of phyllo at a time, fold them accordion-style from the short end and place the ruffled strip in the dish. Repeat until the pan is filled.

Bake the dry crinkled phyllo for 10 minutes. This helps the folds stay crisp after the butter and custard are added.

Pour the melted butter evenly over the phyllo, making sure it runs into the folds. Return the dish to the oven and bake for another 10 minutes.

While the phyllo bakes, whisk together the eggs, 1/2 cup sugar, whole milk, vanilla, and salt until smooth.

Pour the custard evenly over the hot phyllo. Bake for 30 to 40 minutes, until the center is set and the top is deeply golden and crisp.

As soon as the cake comes out of the oven, slowly pour the cooled syrup over the hot pastry. Let it rest for 15 to 20 minutes so the syrup can settle into the layers before slicing into 12 squares.