Heat the olive oil in a large skillet over medium low heat. Add the shallots and salt. Cook, stirring periodically, for several minutes until the shallots are soft and lightly caramelized. Stir in the garlic and cook for another minute. Add the broccoli and cook for another 1-2 minutes until the broccoli has deepened in color and softened slightly. Stir in a few twists of fresh ground black pepper. Move the mixture to a bowl and allow it to cool to room temperature (you can place it in the refrigerator to speed up the process).
Place oven racks in the upper and lower middle positions and preheat the oven to 400 degrees F. Line 2 baking sheets with parchment paper.
Lightly dust a clean surface with flour and roll out one chilled sheet of puff pastry to approximately 12x12 inches (keep the other sheet in the refrigerator until the last second). You will want to work swiftly with the dough before it becomes too soft. Slice into 4 even pieces with a pizza cutter, bench scraper or knife.
Place 3 tablespoons broccoli filling inside one of the squares and top with 1 tablespoon feta cheese. Fold the dough over from corner to corner into a triangle shape, gently pressing the dough against the filling. Slice off some off the excess dough from each side and then use a fork to press the edges firmly together and seal in the filling. Repeat this process with all 4 pieces of dough.
Place the turnovers on a baking sheet and place in the refrigerator for at least 15 minutes. Repeat the same process with the remaining sheet of puff pastry and the remainder of the filling, placing the turnovers onto the second prepared baking sheet.
Whisk the egg together with a splash of water and a pinch of salt.
Remove the baking sheets from the refrigerator and use a small knife to poke 3 holes in the top of each turnover. Use a pastry brush to spread a thin layer of egg wash over each turnover.
Bake for 20-25 minutes or until the dough is cooked through and golden brown on top.
I recommend using good quality feta that has been packed in brine. Make sure to drain it very well before adding it to the turnovers to avoid creating excess moisture.If you have one large baking sheet you can use that instead of two regular baking sheets. If using one large baking sheet, place the oven rack in the middle position.