Put oven rack in middle position and preheat oven to 350°F.
Toss bread cubes with butter in a shallow baking pan, then bake until golden and crisp, 10 to 15 minutes.
Sprinkle with salt.
Stir together parsley and garlic in a small bowl. Coarsely grate cheese (1 cup).
Beat together 3 eggs, 1 teaspoon water, scant 1/4 teaspoon salt, and 1/8 teaspoon pepper in a bowl with a fork until combined.
Heat omelet pan over moderately high heat until hot, then add 1 tablespoon butter and heat, swirling pan, until foam subsides and begins to brown near edge of pan.
Pour beaten eggs into skillet, then cook, shaking pan back and forth quickly with one hand while stirring eggs in a circular pattern with a heatproof rubber spatula with the other hand, until eggs begin to set. Quickly spread set eggs evenly in skillet and sprinkle with 1/4 cup cheese and 1 tablespoon parsley-garlic mixture, then cook until omelet is just set, about 30 seconds to 1 minute.
Add one fourth of croutons (about 1/3 cup) across center of omelet. Holding handle of pan near you, lift edge of omelet closest to handle using a rubber spatula and fold one third of omelet over croutons. Grabbing handle from underneath, tilt pan over plate until unfolded part of omelet slides onto plate, then immediately invert skillet, as if trying to cover plate, to make omelet fold over itself onto plate. Make 3 more omelets in same manner.
