Preheat the oven to 350°. Put the eggs in a saucepan of water and bring to a simmer over moderately high heat. Simmer for 6 minutes. Drain the saucepan and fill it with cold water. Crack the eggs all over and let stand in the water for 1 minute. Peel and thickly slice the eggs; the yolks will be barely cooked but not runny.
Spread the bread pieces on a baking sheet and brush lightly with olive oil. Bake for about 12 minutes, until crisp.
Meanwhile, in a large pot of boiling salted water, cook the asparagus until just tender, about 3 minutes. Drain, cool and cut the asparagus in half lengthwise.
In a small bowl, combine the 1/4 cup plus 2 tablespoons of oil with the vinegar; season with salt and pepper. In a large bowl, toss the asparagus, toasted bread, greens, onion and cheese. Drizzle with the dressing and toss. Garnish with the eggs and radish and serve.