Make the panini: Heat a panini press to high heat.
Spread 1 tablespoon of pesto on each slice of bread. Layer 3–4 Brie slices, 4–5 peach slices, and a drizzle of aged balsamic on 2 slices of bread. Place the other slices of bread, pesto-side down, on top.
Spread 1 tablespoon of butter on one side of each sandwich.
Grease the panini press lightly with nonstick spray. Place a sandwich on the panini press, buttered-side down. Butter the top slice bread, then close the panini press. Cook until the cheese is melted, 2–3 minutes. Remove the panini from the press and repeat with the remaining sandwich.
Make the salad: Arrange the peach wedges on the panini press and cook until grill marks appear, 2–3 minutes. Remove from the press.
In a medium bowl, drizzle the arugula with olive oil and aged balsamic vinegar. Toss well. Add the grilled peaches.