Preheat oven to 350°.
Heat 1 tablespoon oil in a large ovenproof skillet over medium heat. Add 1 garlic half to pan; cook 1 minute. Add bread cubes; toss to coat. Bake at 350° for 10 minutes or until browned, stirring occasionally.
Place eggs in a medium saucepan. Cover with cold water to 1 inch above eggs. Remove eggs from pan; bring water to a boil. Using a spoon, gently lower eggs into boiling water. Cover saucepan, remove from heat, and let stand 6 minutes. Place eggs in a bowl of ice water, and cool completely. Gently crack eggshells, and peel over a large bowl, pouring yolks into bowl. Coarsely chop egg whites. Discard eggshells.
Mince remaining garlic. Add minced garlic, lemon juice, and next 3 ingredients (through anchovies) to yolks; mash with a fork to form a paste. Gradually add remaining 3 tablespoons oil, 1 tablespoon at a time, stirring well with a whisk until each addition is incorporated and mixture emulsifies and thickens slightly. Add chopped egg whites, croutons, arugula, and mint to dressing. Sprinkle evenly with pepper and salt; toss gently. Sprinkle with cheese. Serve immediately.