Preheat the oven to 350 degrees F (175 degrees C). Butter a 9x13x2-inch baking pan.
Drain strawberries and reserve juice in a bowl. Chop into pieces and add any remaining juice to the bowl.
Combine cottage cheese, cream cheese, 1/2 cup sugar, egg yolks, whole egg, vanilla extract, and lemon zest in a bowl. Blend together using an electric mixer on medium speed. Stir in the drained strawberries.
Place phyllo dough sheets between dampened towels to keep moist.
Place 1 phyllo sheet on the prepared pan. Brush with melted butter. Repeat until 1/2 of the sheets are left.
Spoon the strawberry-cheese mixture over the top layer of phyllo; spread evenly. Layer the remaining phyllo sheets on top, brushing each with butter.
Bake in the preheated oven until golden brown, about 1 hour.
Combine 1/2 cup reserved strawberry juice, remaining 1/2 cup sugar, and lemon juice in a saucepan over medium heat. Cook into a thick syrup, about 5 minutes.
Spoon hot syrup evenly over the baked baklava. Let cool before cutting into pieces.
