To make the Bois Boudran Sauce: Place the vinegar in a large bowl. Gradually add the oil, whisking constantly until well blended. Whisk in the mustard and Worcestershire sauce. Whisk in the ketchup. Stir in the chives and tarragon. Set the sauce aside.
To prepare the Burgers: Heat 1 tablespoon/15 ml of oil in a heavy small sauté pan over medium heat. Add the shallots. Sprinkle with salt and pepper and sauté for 3 minutes or until soft. Stir in the garlic. Remove from the heat and allow to cool.
Gently mix the ground beef, parsley, mustard, Worcestershire sauce, salt, pepper, sautéed shallots and garlic in a large bowl until well blended. Mix in the egg.
Divide the beef mixture into 4 equal mounds and shape each mound into a 4-inch/10-cm diameter patty. Set the patties aside. Lay the bacon strips in a heavy large frying pan and cook over medium heat for about 8 minutes or until crisp and golden.
Remove from the heat and set the bacon aside. Let rest for 1 minute. Transfer the cooked bacon to paper towels to absorb the excess oil. Break the bacon strips in half. Broil the brioche buns cut side up for 1 to 2 minutes or until lightly toasted.
Meanwhile, heat a heavy large nonstick griddle pan or sauté pan over medium-high heat. Drizzle the pan with 1 tablespoon/15 ml of olive oil and cook the beef patties for 2 to 3 minutes or until brown on the bottom.
Turn the patties over. Top the patties with the Gruyere cheese and cook for about 3 minutes longer or until the patties are cooked to medium-rare doneness.
Set the patties on the bun bottoms. Place the crispy bacon on top of the patties. Sauté the mushrooms and place the mushrooms on top of the bacon on the patties. Cover with the bun tops. Serve with the Bois Boudran sauce and enjoy!