Divide the broccoli into very small florets, cut the bell pepper into small cubes, mushrooms into slices.
Heat a tablespoon of oil in a large skillet over high heat. Add the broccoli and bell pepper in an even layer. Don't stir for a minute, then cook, stirring from time to time for a few minutes, until the vegetables are soft and browned. Add the herbs and garlic, cook for another minute. Season to taste with salt and pepper. Transfer to a large chopping board.
Add a second tablespoon of oil and mushrooms to the pan. Cook just like the vegetables. Transfer to the chopping board.
Chop the vegetables and mushrooms. Set aside to cool completely. Once cooled, stir in the cheese. Place the bowl with the filling in the fridge.
Heat the oven to 400°F / 200°C / Gas Mark 6, no fan (if you have a convection oven, reduce the temperature by 20° or follow the manufacturer's instructions).
Sprinkle the puff pastry well with flour and roll out on a sheet of parchment paper into a long rectangle. Place the filling on its half. Cover the filling with the second part of the dough and press the edges of the dough together to seal.
Cut the dough horizontally with a sharp knife into parts (not just score, but cut it all the way, it will be easier to divide the pastry into pieces after baking it, you can also just score the dough if that's what you prefer).
Make the egg wash - whisk the egg with milk. Brush the pastry with the mixture.
Bake for about 25 minutes or until the pastry is golden. Wait 10-15 minutes before dividing it into parts.