Toss the peaches with 2 teaspoons of the oil, balsamic vinegar, and a pinch of salt. Heat a grill or grill pan to medium and brush with oil. Place the peach wedges on the grill, and cook undisturned for 2-3 minutes on each side, until grill marks appear and peaches are heated through. Transfer to a dish and allow to cool to room temperature.
Combine the mint and basil in the bowl of a food proccessor. Add the 1/4 cup of olive oil, toasted pine nuts, chopped garlic, lemon juice, and lemon zest, a pinch of red pepper flakes, and salt and pepper. Pulse until smooth. If the pesto seems too dry, add a bit of water, a teaspoon at a time.
Toss the arugula with a drizzle of olive oil and arrange on a serving platter. Top with the grilled peaches, dollops of the mint pesto, and serve.