Preheat the oven to 375 degrees F. Line a baking sheet with parchment paper. Heat the olive oil in a large skillet over medium-high heat. When the oil shimmers, add the onion and cook 5 minutes, until softened. Add the broccoli, mushrooms (if using), and season with salt and pepper. Continue cooking another 5-10 minutes or until the broccoli begins to char. Add the garlic, thyme, nutmeg, and cayenne, cook another minute. Remove from the heat.Stir in the cream cheese, 1 cup cheddar, and Havarti cheese.Lay the pastry sheets flat on a clean, floured surface. Gently roll the sheets out to stretch slightly. Cut each sheet into 9 squares. Evenly divide the broccoli filling among half of the squares, leaving a 1/4 inch border around the edges. Brush the edges with egg. Lay the remaining 9 squares over the filling and seal the edges by crimping with the back of a fork. Place the squares on the prepared baking sheet and brush the tops with beaten egg. Make 2 small slits on the top of the pastries. Sprinkle with the remaining 1/2 cup cheddar cheese and the everything spice, if desired. Transfer to the oven and bake for 20-25 minutes or until golden brown.Serve warm, topped with additional thyme.