Light a grill. In a medium bowl, toss the peaches with 2 tablespoons of the olive oil and the thyme and season with salt and pepper. Let stand for 10 minutes.
Grill the peach halves over high heat for about 6 minutes, turning once, until lightly charred and softened.
Meanwhile, in a small skillet, toast the pine nuts over moderate heat, stirring, until golden, about 3-4 minutes. Transfer the nuts to a cutting board and finely chop.
Put the pine nuts in a medium bowl. Whisk in the vinegar and the remaining 2 tablespoons of olive oil and season with salt and pepper. Add the arugula and toss. Arrange the goat cheese slices on plates. Top with arugula salad and then grilled peaches and serve right away.