Make the dough: combine the water, olive oil, lemon juice, and kosher salt in a large mixing bowl. Add 3 cups of the flour all at once, and stir with your hands until a ball forms. Remove from the bowl and transfer to a floured surface, kneading and incorporating up to ¾ cup more flour until the dough is no longer sticky.

Divide the dough into 20 even pieces (around 40-45 grams each) and roll into balls. Use a rolling pin to roll the balls into flat, thin circles (⅛” thick) and dip each one into the remaining 1 cup of oil, ensuring both sides are covered. Stack the dough circles on a large, deep plate. Let your dough rest for an hour, covered.

In the meantime, prepare the filling. Heat 1 tablespoon of the olive oil in a saucepan over medium heat. Add the onion to the pan and saute until fragrant, 2-3 minutes. Add the rest of the olive oil, the eggplant and the plum tomatoes to the pan, stirring to combine. Season with the sea salt and black pepper and cook over low heat, stirring as needed, until the mixture is reduced, 30-35 minutes. Remove from the heat and add ½ cup of the Parmesan.

Preheat the oven to 375°F. Using your hands, take each dough circle and stretch it thin on your work surface until you can see through the dough. Don’t worry if the dough tears slightly. Once your dough is see through, use a pizza cutter or a sharp knife to cut away the edges, leaving a square.

Fold your square of dough into thirds. Then fold into thirds again, the opposite way. Place one heaping teaspoon of filling in the middle of your dough, then bring opposite corners over your dough to make a little envelope. Repeat with remaining dough.

Place the boyos on a parchment paper lined baking sheet. Brush the filled boyos with the egg wash, then sprinkle with the remaining Parmesan. Bake for 30-35 minutes or until golden. Remove from the oven and serve warm.