Split croissants horizontally. Spread each half with 1-2 tablespoons of cranberry sauce (depending on how sweet you like your panini), and a few slices of Brie. Layer with turkey and top with baby arugula leaves. Close croissant halves.
Preheat a panini press, or a grill pan on the stove over medium flame until hot. If using a panini press, oil the grates and cook the sandwich according to your manufacturer's directions. If using a grill pan, give it a light coating of cooking spray. Place panini on the grill and weight with a grill press or a small sauté pan weighted with a heavy can. Grill 3-4 minutes per side, until croissants are toasted and cheese is melted. Cut in half and serve.