Step 1Brush the truffle with a damp cloth and scrape away rough spots and dirt with a paring knife. Slice the truffle into very thin rounds with a truffle shaver or Japanese mandolin; stack the rounds and cut crosswise into thin sticks.
Step 2Beat the eggs lightly with half the julienned truffles, one-half teaspoon salt and freshly ground pepper. Melt 1 1/2 tablespoons of the butter in an omelet pan over medium heat. Add the eggs, stirring with a wooden spoon to create soft curds. When the eggs lose their shine, cut the remaining butter into small pieces and stir in. Remove the pan from the flame when the eggs are still a bit too soft; they will continue to cook off the heat.
Step 3Divide the eggs among 4 warmed plates and garnish with the remaining slivered truffle. Quickly dress the salad greens with 2 tablespoons olive oil, lemon juice and salt, and put a handful of dressed greens alongside the eggs on each plate. Serve with toasted levain bread, drizzled or brushed with the remaining olive oil and rubbed with garlic.