In a large pot filled with cold water, soak the corn for 30 minutes.
Blanch the asparagus in a large pot of boiling water and transfer to a bowl of ice water. Meanwhile, prepare the Arugula Pesto.
Prepare a stove-top griddle or outdoor grill. Grill the corn in the husk, turning occasionally, for 15 to 20 minutes. (If your grill has a cover, close it to allow the corn to steam.) Brush the asparagus with some of the olive oil and grill for 4 to 5 minutes, turning once or twice. Use a vegetable tray to prevent the asparagus from falling through the grill rack. Transfer the corn and asparagus to a large platter and serve with the Arugula Pesto.
Grill the crabs until bright orange and slightly charred, about 2 to 3 minutes per side. Season with salt and pepper. Place the greens in a large salad bowl and set the crabs on top.