In a small bowl, whisk 3 tablespoons of the olive oil together with the red wine vinegar and mustard. Season with salt and pepper to taste and set aside.
Heat an indoor or outdoor grill over medium high heat. Toss the asparagus and squash with the remaining olive oil and season with salt and pepper. Transfer the vegetables to the grill and grill for 5-6 minutes until the asparagus is just tender and the squash has grill marks. Remove from the grill using a pair of tongs and set aside. Cut the asparagus into 2 inch pieces.
Toss together the dressing and the arugula in a large bowl and season with salt and pepper as needed. Add the squash and the asparagus and toss to combine.
Finally, heat the butter in a nonstick skillet over medium heat. Cook the eggs, sunny side up, until the white part is firm but the yolk still running, about 4-5 minutes and season with salt and pepper.
Divide the salad evenly amount 4 plates, top each with a fried egg and serve immediately.