A favorite at L.A. restaurant Campanile, this recipe is adapted from "Nancy Silverton's Sandwich Book: The Best Sandwiches Ever: From Thursday Nights at Campanile."
Marinate anchovies: In a small bowl, combine anchovy fillets, oil, lemon zest, and parsley. Toss to combine. Let stand at room temperature, 1 hour.
Prepare an ice bath; set aside. Place eggs in a medium saucepan. Add enough water to cover. Bring to a boil over high heat. Reduce to a low simmer, and cook, 6 to 7 minutes. Transfer to ice bath until cool. Peel, and cut into quarters, set aside.
Heat a grill pan over medium-high heat. Brush both sides of bread with clarified butter. Place in skillet and cook until golden, 2 to 3 minutes per side. Rub one side of each slice with garlic, and place on work surface, garlic-side up.
Spread about 2 tablespoons aioli unevenly over the bread. Cut each slice of bread in half on the diagonal and place on serving plates.
Cut braised leeks in half on an angle, and divide them, fanning slightly, between slices of bread. Place 2 quarters of egg over each leek. Drape an anchovy fillet over each egg quarter, and drizzle with some of the anchovy marinade.