Preheat the oven to 375 degrees F.
Pat the chicken dry with paper towels. Rub the exterior and cavity liberally with the olive oil. Sprinkle with salt and pepper. Stuff the cavity with the garlic, carrots, celery stalk, lemon, onion and thyme. Tie the legs tightly with kitchen twine and fold the wing tips under the breasts. Place on a roasting rack in a deep pan and into oven. Roast until a digital read thermometer registers 165 degrees F and the juices run clear, about 1 1/2 hours. Remove and cool.
Once cooled enough to handle, remove the meat from the bones. Shred against the grain with two forks. Cool completely.
Whisk the mayonnaise, cream cheese and lemon juice in a large mixing bowl for about 30 seconds. Add the chicken, diced celery, grapes and apples. Toss to evenly coat. Fold in the walnuts, parsley and tarragon until thoroughly incorporated. Season with salt and pepper.
To assemble, use a preheated panini press on the highest setting. If unavailable, bring the oven to 400 degrees F and use two cast-iron skillets to form a press.
Cut the focaccia quarters in half with a sharp serrated knife and arrange on a prep surface. Spoon about 1 1/4 cups of the chicken salad onto each bottom half. Replace the top half. Place onto the press and crisp until the bread is golden on both sides, 3 to 5 minutes. If using skillets, place the sandwich into the bottom pan, place the second skillet bottom on top of the sandwich and crisp, flipping halfway through the cooking process.
To serve: Remove the sandwiches from the press or oven to a prep surface. Cut in half diagonally with a sharp serrated knife and arrange on serving plates. Enjoy immediately.