Whisk together the balsamic vinegar, sorghum syrup, apple cider vinegar, smoked paprika and some salt and pepper in a medium bowl until smooth. Slowly whisk in the oil until emulsified. Set aside.
Heat a medium nonstick skillet over medium heat. Brush some of the sorghum vinaigrette on the cut sides of the peaches and put them, cut-side down, in the hot skillet. Cook just until the vinaigrette caramelizes, about 2 minutes.
Transfer the peaches, cut-side up, to a serving platter and arrange slices of burrata around them. Scatter the ham over and around the peaches and burrata, then scatter the arugula on top. Drizzle with a bit more sorghum vinaigrette and grind some black pepper over the top. Serve immediately.