Preheat the oven to 425°F. If you have a convection setting, that is best.

In a saute? pan, heat 2 tablespoons of the oil. Sweat the shallots and garlic until they are soft. Add the mushrooms and brown them on all sides. Add more oil or a bit of butter toward the end if the pan becomes too dry. Add the leeks and tomatoes. Cover partially with a lid and cook on low heat until all of the excess liquid has evaporated. Season with salt and pepper.

Cut two large pieces of parchment paper in half. Fold each piece in half and cut off the corners, rounding off each edge, so that when unfolded, all outer and inner corners of each piece of paper are rounded.

Place a fourth of the vegetable mixture on one rounded half of each piece of parchment. Top each with a whole sprig of thyme and moisten with 2 tablespoons of white wine.

Season the quail liberally with salt and pepper. Insert a toothpick through the legs to keep them together. Then lay one quail over each bed of vegetables.

Using a pastry brush, brush the edges of the parchment paper with the beaten egg white, and press the edges together to seal. Brush the edges of the folded package with the beaten egg white and make a series of short folds along the edges. Brush the edges again with egg white and repeat the short folds. Very lightly brush the top of each parchment package with oil.

Place the papillotes on a sheet tray. Bake for 10 minutes, until completely puffed and the parchment has browned. Serve the quail in the papillote to be opened at the table, so that people can enjoy the aroma when they open the package.