For dressing, in a small bowl whisk together vinegar, olive oil, sugar, and the 1 tablespoon chives. Season to taste with salt and pepper; set aside.
Shred cabbage and place in a large bowl. Halve, pit, and thinly slice peaches; add to bowl with cabbage. Gently toss to combine. Drizzle with about half of the dressing; toss to coat. Top with blackberries and crumbled cheese. Sprinkle with herbs; pass remaining dressing.