Make sure the onions are cooked until soft for best flavor with the greens.
Heat grill to medium; lightly oil grates. In a small bowl, gently toss onion with 1 tablespoon oil; season with salt and pepper. Grill wedges, turning occasionally, until soft, 15 to 20 minutes.
Meanwhile, in a large bowl, whisk together lemon juice and remaining 3 tablespoons oil; season with salt and pepper. Add greens, and toss to coat. Serve topped with onion and cheese.
Keeping a bit of the root attached to the wedges prevents the onion layers from falling apart.
