Combine the first five ingredients for the vinaigrette into a blender and blend on high. While the blender is still going on high, slowly add the oil. Place the vinaigrette into a glass jar or container and set aside.
To Grill the Veggies:
Preheat the grill to 350-375 degrees. Place the squash, zucchini, asparagus, and onion in a large sheet pan. Place the radishes into the small aluminum pan. Drizzle the veggies with olive oil and season with salt and pepper.
Place the onion cut side down and the pan with radishes on the grill and cook for about 15 minutes stirring the radishes occasionally. Then place the zucchini and squash cut side down on the grill. Cook for about 10 minutes, turning once. Add the asparagus to the grill and cook for 5-7 minutes or until tender. The veggies should be tender but still have a bite, cooking times will vary depending on the size of the veggies. Remove the vegetables from the grill and set aside for about 5 minutes to cool.
To Prepare the Salad:
Dice the squash and zucchini and place in a medium mixing bowl. Cut the radishes in half and add to the zucchini. Cut the asparagus into 2-inch pieces and cut the onion into large dice or wedges and add them to the bowl. Add about half of the prepared dressing and the shaved Parmesan and toss to coat. Taste and adjust salt and pepper or add more dressing as needed. Serve warm or at room temperature.