Preheat the oven to 450 degrees . In a large skillet, heat 3 tablespoons olive oil over medium-high heat. Add the mushrooms, season with salt and pepper and cook until the liquid evaporates, about 10 minutes. Transfer to a bowl along with the parsley, scallions and lemon peel; season with salt and pepper. Stir in the cream cheese and 1 teaspoon lemon juice.
On a lightly floured surface, roll out 1 puff pastry sheet to an 11-inch square, then halve it. Repeat with the other pastry sheet. Arrange one-quarter of the filling crosswise on 1 half of each piece of pastry, leaving a 1/2-inch border. Brush the borders with some egg, then fold the pastry to enclose the filling. Cut 2 vents into each turnover and crimp the edges; transfer to a parchment-paper-lined baking sheet. Brush the tops with more egg. Bake until puffed and golden-brown, about 18 minutes. Let rest for 5 minutes.
In a serving bowl, whisk together the remaining lemon juice and 1 tablespoon olive oil; season with salt and pepper. Add the spinach and toss. Serve with the turnovers.