To make the vinaigrette, pour the vinegar into a small bowl and whisk in 5 tablespoons of the extra virgin olive oil. Season to taste with salt and pepper. Set aside.
Cut the carrots crosswise in half, then cut the halves into thick rough sticks. Toss in a bowl with 3 tablespoons of the olive oil, the thyme, and salt and pepper to taste.
Heat a large cast-iron skillet over high heat. Working in batches if necessary, add the carrots in a single layer and cook, without turning, until they are charred on the bottom and almost burned, 3 to 5 minutes. Turn with a spatula and cook on the other side for 2 to 3 minutes more, adjusting the heat as necessary, until they are crunchy on the outside and tender within. Transfer to a tray.
Combine the parsley and arugula on a large serving platter and toss lightly with half the vinaigrette. Place the carrots on top.
Toss the garlic chips over the salad and drizzle with the remaining vinaigrette.