1. Adjust oven rack to highest position and heat broiler. Brush bread with about 2 tablespoons of the oil; broil bread, turning once, until toasted on both sides. Rub toasted bread with garlic, then break each piece into large pieces to make croutons.
2. Mash the same garlic clove with 1/4 teaspoon salt in small bowl until smooth; whisk in vinegar, then remaining oil, and finally shallots.
3. Heat this vinaigrette in small saucepan until hot. Pour hot dressing over greens in large serving bowl; toss lightly to coat. Add croutons and plenty of pepper; toss again. Arrange egg quarters and cheese shavings over greens and serve immediately.