Whisk together olive oil, honey, lime juice, vinegar, Dijon, and a pinch of salt and pepper in a small bowl or jar until emulsified. Taste and adjust sweetness or acidity.
Heat a small skillet over medium heat. Add the almonds or pistachios and toast, stirring frequently, until fragrant and lightly golden, about 2–3 minutes. Transfer to a plate to cool.
Halve and pit the nectarines, then slice into thin crescents. If fruit is very soft, handle gently to avoid mashing.
On a large platter or individual plates, layer arugula, nectarine slices, torn burrata, and red onion if using. Drizzle about two-thirds of the dressing over the salad and gently toss or arrange for presentation.
Sprinkle toasted nuts and torn basil on top. Season with flaky sea salt and cracked black pepper. Serve immediately with extra dressing on the side.
Notes
For a smoky flavor, quickly char nectarine halves on a hot grill for 30–60 seconds per side before slicing.
Swap burrata for fresh mozzarella or goat cheese if preferred.
Make the vinaigrette ahead and store in the fridge for up to 3 days; bring to room temperature and whisk before serving.