Preheat the oven to 400 degrees F. Place a skillet over medium heat. Add the butter, sliced shallots and mushrooms. Saute for 5 minutes to brown the mushrooms. Then salt and pepper to taste, and add the wine. Allow the mushrooms to simmer until the wine has completely absorbed and the mushrooms are nearly dry, about 15-20 minutes.
Meanwhile, cut one sheet of puff pastry into 4 squares and place them on a parchment paper-lined baking sheet. Top each square with 1/4 cup of cooked rice, make sure to leave a border around the edges.
Squeeze the spinach to remove any excess liquid, then layer the spinach over the rice. Sprinkle the spinach with a touch of salt. Once the mushrooms have cooked down, layer them on top of the spinach. Top each stack with a half-portion of salmon. Salt and pepper the salmon and sprinkle with fresh dill.
Whisk the egg with 1 tablespoon water and brush it around the exposed puff pastry edges.
Roll the second piece of puff pastry out into a slightly larger square. Cut it into four smaller squares. Lay each square over a salmon stack. Gently seal the edges by crimping with a fork.
Brush the remaining egg over the tops. Cut a small vent in the top of each pastry. Bake for 20-30 minutes until the tops are golden.