Make the dressing. In a small bowl or jar, whisk together olive oil, lemon juice, honey, Dijon, salt, and pepper until smooth and slightly thick.
Taste and adjust with a bit more lemon or salt if needed.
Prep the peaches. Slice peaches into thin wedges. If they’re very juicy, set them on a paper towel for a minute so they don’t water down the salad.
Toast the nuts. Warm a dry skillet over medium heat. Add nuts and toast for 3–4 minutes, shaking the pan often, until fragrant and lightly golden.
Cool slightly, then chop.
Build the base. Add arugula to a large bowl. Drizzle with 1–2 tablespoons of dressing and toss gently to lightly coat. This keeps the greens crisp and evenly seasoned.
Add toppings. Scatter peaches, cheese, red onion, basil, and toasted nuts over the arugula.
Dress and finish. Drizzle on more dressing to taste.
Season with a pinch of salt and several grinds of black pepper. Toss lightly just before serving so the peaches and cheese don’t break down.
Serve. Plate immediately while everything is fresh and vibrant.