Preheat oven to 400°F (200°C). Toss bread cubes with olive oil and bake for 10-15 minutes until golden and crispy. Set aside.
Blanch the asparagus and peas in boiling water for 1-2 minutes, then transfer to an ice bath to stop cooking.
In a large mixing bowl, combine the toasted bread, asparagus, peas, radishes, cherry tomatoes, arugula, red onion, and basil.
Whisk together all dressing ingredients in a small bowl or jar until well combined.
Pour dressing over the salad and toss to combine. Let sit for 10 minutes before serving to allow flavors to meld.
Top with shaved Parmesan and serve immediately.