Slice the zucchini lengthwise into 1⁄3-inch-wide strips.
Preheat a grill pan over high heat for 5 minutes. In a large bowl, toss the zucchini strips with 1 tablespoon of the oil, 1⁄2 teaspoon salt, and a grind or two of pepper. Working in batches, grill the zucchini for 4 to 5 minutes, turn and grill until tender and well marked, 4 to 5 minutes more. Remove the zucchini from the heat, mound it on a baking sheet to gently finish cooking, and let sit for 10 minutes.
Whisk together the vinegar and remaining 3 tablespoons oil. Transfer the zucchini to a platter. Drizzle with the vinaigrette, sprinkle with the pistachios and mint, toss lightly, season to taste with salt, garnish with arugula, and serve.