Preheat the oven to 450°. 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.
Toss the filling with the baby spinach before sealing inside the pastry.
Freeze uncooked turnovers for up to 1 month. Let stand at room temperature for 20 minutes before baking as directed.