Make the turnovers: Preheat the oven to 375 degrees F. Melt the butter in a large saute pan over medium heat. Add the onion, carrot and celery, season with salt and pepper to taste, and cook until the carrot softens, about 10 minutes. Add the garlic and thyme and cook for an additional minute, stirring frequently. Add the flour and cook off the raw flavor, about 1 minute more. Turn up the heat to medium high and deglaze the pan with the wine. Whisk in the stock and dijon mustard. Add the peas and chicken and cook at a very low simmer until the sauce thickens up into a gravy, 5 to 10 minutes. Remove from the heat and set aside.
On a floured surface, unfold the puff pastry and cut into 4 squares (cut once horizontally and once vertically). Gently roll out each square, increasing the size of the square by about 20 percent. Place on a baking sheet and spoon the chicken mixture in the center. Fold the squares in half diagonally to create triangles. Pinch the edges together to create turnovers (use a little water if needed to bind). Press the edges with a fork to create a decorative border. Whisk the egg with a splash of water and brush the turnovers with the egg wash. Cut a small slit in the tops to allow the steam to escape during baking. Bake the turnovers until golden brown, about 30 minutes. Let cool at least 10 minutes before serving as the insides will be very hot.
Meanwhile, make the salad: Rub the inside of a large wooden bowl with the garlic clove and then either discard it or save it for another use. Put the mustard in the bowl and whisk in the vinegar and soy sauce (a French secret!) vigorously for about 10 seconds to get a creamy consistency. Season with salt and pepper. Drizzle in the olive oil as slowly as possible with one hand while whisking as quickly as possible with the other hand to emulsify. Lay the salad greens on top of the dressing and toss just before serving. Serve with the turnovers.