Brush all sides of the peaches lightly with 1 Tbsp. (or less) of olive oil. Grill the peaches over a medium fire until they brown lightly and begin to soften slightly, 4 to 6 minutes on each side. (Or you can grill the peaches on the stove in a grill pan!) Transfer the grilled peaches to a platter and let them cool slightly. When cool enough to touch, cut each half into 1/2-inch-thick wedges.

Meanwhile, in a small bowl, whisk together the balsamic vinegar and olive oil, along with a pinch of salt and a few grinds of fresh pepper to taste. Then add the arugula to a large bowl, and toss it with the vinaigrette. Once the arugula is evenly coated, divide the greens among 4 salad plates. Top each salad with a sprinkling of cheese and pecans, then arrange 4 or 5 grilled peach wedges atop the greens. Serve immediately.