In a medium skillet over medium heat, melt butter and olive oil.  Add smashed garlic cloves and the whole sprigs of rosemary.  Sauté until fragrant, about 2 to 3 minutes.  Add mushrooms, salt, and pepper.  Sauté about 10 to 12 minutes, stirring often, until all of the liquid has been absorbed and the mushrooms are tender.  Remove garlic cloves and rosemary.  Set aside.

Spoon the mushroom mixture onto one of the rectangles, leaving a 1-inch border all around.  Using a pastry brush, coat the 1-inch border of pastry with beaten egg.  Lay the remaining rectangle of pastry over the filling and press gently to seal.  Leave the pastry on the parchment-lined baking sheet and refrigerate another 20 minutes (or up to 2 hours).

Preheat oven to 375 degrees F.

Using a small knife, trim the edges of the pastry. Using a pastry brush, coat the entire top of the pastry with the beaten egg.  Gently cut five 3-inch vents in the top, being careful not to pierce the bottom layer.  Bake, rotating once, for about 35 to 40 minutes, until the crust is deep golden brown and the pastry has puffed.  Using one or two spatulas, transfer the pie to a wire rack and let cool for about 10 minutes before slicing and serving.  The pie also tastes great at room temperature.

Spoon the pesto onto one of the rectangles, leaving a 1-inch border all around.  Sprinkle with salt and pepper. Top with slices of mozzarella. Place the tomato slices on top of the cheese. Using a pastry brush, coat the 1-inch border of pastry with beaten egg.  Lay the remaining rectangle of pastry over the filling and press gently to seal.  Leave the pastry on the parchment-lined baking sheet and refrigerate another 20 minutes (or up to 2 hours).

Preheat oven to 375 degrees F.

Using a small knife, trim the edges of the pastry. Using a pastry brush, coat the entire top of the pastry with the beaten egg. Gently cut five 3-inch vents in the top, being careful not to pierce the bottom layer.  Bake, rotating once, for about 35 to 40 minutes, until the crust is deep golden brown and the pastry has puffed.  Using one or two spatulas, transfer the pie to a wire rack and let cool for about 10 minutes before slicing and serving.  The pie also tastes great at room temperature.