Preheat a panini grill or sandwich press.
Lay each sliced baguette square on a work surface with the bottoms of the buns side by side.
Divide half of the Parmesan among the 4 bottom buns.
Then equally divide the turkey or chicken, red peppers, mushrooms, olives, and arugula among them as well and make even layers over the cheese.
Drizzle a few drops of olive oil over the arugula on each sandwich and sprinkle the thyme leaves over as well.
Season to taste with salt.
Top with the remaining Parmesan and return the tops of the bread to the sandwiches.
Lightly oil the outside pieces of bread with extra-virgin olive oil.
Press firmly to secure buns and working in batches of two, grill the sandwiches in the panini press for 5 to 6 minutes or until the bread is golden brown and the cheese has melted.
Once the paninis come off the grill, rub the grilled side of the bread lightly with the garlic clove.
Slice the sandwiches in half diagonally and serve warm.