Loosely cover the folded sheets of frozen puff pastry with a clean kitchen towel and thaw at room temperature for 45 minutes or until pliable enough to work without cracking.
Place an 11-inch skillet over medium-high heat for 1 minute. Add the mushrooms and 1/4 cup of water and cook until the mushrooms collapse and most of the water evaporates, 2-3 minutes. Add the butter and continue cooking, stirring frequently until the mushrooms begin to brown, about 4 minutes. Remove to a large mixing bowl and season with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper.
Set your oven racks to the upper and lower thirds of the oven and crank heat to 400℉.
Lightly dust a work surface with flour and gently unfold one of the pastry sheets. Roll the dough to an 11 x 11-inch square, using a rolling pin. Pierce the dough several times over the entire surface with a fork (this helps in preventing the dough from overly puffing while baking). Cut into 4 equal squares using a pizza wheel or sharp knife, and place on a sheet pan lined with parchment paper. Repeat the process with the remaining pastry sheet and place on a second parchment-lined sheet pan.
Lightly beat one of the eggs and add it to the bowl with the mushrooms, followed by the salmon, cornichons and their juice, the Dijon mustard, scallions, parsley, and the remaining salt and pepper. Thoroughly combine with a spoon or silicone spatula, breaking up any clumps of salmon.
Beat the remaining egg in a small bowl with 2 tablespoons of water, then use a brush to apply a thin layer to the edges of each of the dough.
Place equal amounts of the filling in the center of each square and fold over to make a triangle. Use a fork to crimp and seal the edges and to pierce several small holes on the top to all steam to escape. Brush the turnovers with more of the egg wash (for browning).
Bake for 15 minutes then reverse the pans (top to bottom rack and vice versa), and bake for another 15 minutes or until golden brown. Cool for 15 minutes before serving.