Place the chopped pecans in a dry skillet over medium heat and toast for 3 to 5 minutes, stirring often, until they smell nutty and darken slightly. Tip them onto a plate right away so they stop cooking.
In a small bowl or jar, combine the balsamic vinegar, 2 tablespoons olive oil, honey, and Dijon mustard. Add the salt and a few grinds of black pepper, then whisk or shake hard until smooth and slightly thickened. Taste and adjust sweetness or sharpness as needed.
Cut each peach in half along the natural seam, twist to separate, and remove the pits. Brush the cut sides generously with the 2 tablespoons of olive oil.
Set your grill pan over medium-high heat, or heat an outdoor grill to about 400°F (200°C). Let the pan get fully hot before the peaches go on, about 3 to 4 minutes.
Lay the peaches cut side down on the hot grill and cook undisturbed for 3 to 4 minutes, until deep grill marks form and the sugars caramelize. Turn with tongs and grill the skin side for 1 to 2 minutes, just to warm through, then transfer to a plate.
While the peaches rest, wash the arugula and dry it well in the salad spinner. Add the greens to your large bowl along with the thinly sliced red onion, drizzle in about two-thirds of the vinaigrette, and toss gently to coat.
Cut each grilled peach half into thick wedges, or leave them as halves for a more dramatic look. Spread the dressed greens on a platter or divide among plates and arrange the warm peaches on top.
Scatter the crumbled goat cheese and toasted pecans over the salad. Drape the torn prosciutto around the peaches if using, tear the fresh basil over the top, and drizzle with the remaining vinaigrette. Finish with a pinch of flaky salt and serve right away while the peaches are still warm.