In a small bowl, whisk together the vinegar, garlic, mustard and olive oil. Season to taste with salt and pepper and set aside.

Cut each length of French bread in half horizontally. Using a pastry brush, brush the cut side of each piece of bread with some of the vinaigrette. Set bread aside.

Add the bacon to a small skillet and cook over medium-high heat until the bacon has rendered its fat and is crispy. Remove the bacon using a slotted spoon and set aside on paper towels to drain. Add the mushrooms to the bacon drippings remaining in the skillet and season with salt and pepper. Cook, tossing occasionally, until the mushrooms are golden brown and tender, 4 to 6 minutes. Remove from the heat and set aside.

Preheat a panini press or a grill pan to medium heat and lightly oil the grates if necessary.

To assemble the sandwiches, divide the turkey evenly among 4 pieces of the bread, then top each with even portions of the mushrooms, bacon, and cheese. Sprinkle with the herbs, then place the untopped pieces of bread on top to close the sandwiches. Brush the outside of the sandwiches lightly with olive oil.

Cook the sandwiches, in batches if necessary, pressing with the top of the panini press or weighted with a heavy pan if using a grilling pan, until the bottoms are crispy and golden. Turn if necessary to crisp the second side of the sandwiches, and cook until the cheese is completely melted and the sandwiches are warmed through.

Cut the sandwiches in half and serve hot.