Cut each ciabatta loaf in half. Brush the inside of each half with olive oil and season with salt and pepper. Place on a baking sheet and set aside.
Prepare the grill and let it burn down to medium-hot coals or set a gas grill to medium heat. Place the bread on the grill, away from direct heat, and grill on both sides until brown and crispy on the outside but still soft within, about 5 minutes.
Leave the mushrooms whole, and toss in a bowl with 1/2 cup of the olive oil, 1 teaspoon of the thyme, the garlic, and salt and pepper to taste. Immediately grill the mushrooms (be careful not to allow the flames to flare up and burn the mushrooms), turning at least once, until browned and cooked through, about 7 to 8 minutes.
In the marinade bowl, add the shallot, 1/2 cup of olive oil, vinegar, parsley, the remaining 1 teaspoon thyme, and salt and pepper, to taste. Stir well. Let the mushrooms cool, finely chop, and add to the marinade bowl. Toss well. (The mushroom vinaigrette may be made a day ahead, covered, and refrigerated, and then brought to room temperature before serving.)
Sprinkle salt and pepper on both sides of the chicken. Drizzle with a little olive oil.
When the coals have burned down to a medium fire, place the chicken on the grill and cook until browned, about 2 minutes. Turn over, move to the edge of the grill away from direct heat, and continue to cook until opaque throughout, about 7 minutes. Remove to a plate.
In a bowl, toss the arugula with the citrus vinaigrette. Slice the chicken breasts crosswise at a 45 degree angle. Pour any accumulated chicken juices into the mushrooms and stir. Spoon the mushrooms on top of 4 slices of grilled bread and top with the chicken. Divide the arugula among the sandwiches. Place remaining slices of bread on top and serve immediately.
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