It doesn't get more Italian-American than this!
In small bowl, mix mayonnaise, scallions, and oregano; season. In large nonstick skillet, cook sausage patties over medium-high until cooked through, 3 to 4 minutes per side. Transfer to plate. Wipe out any blackened pieces in skillet. Fry eggs over medium-low until whites are set, about 4 minutes. Place cheese on cut sides of bread. Broil until melted, about 2 minutes. In bowl, toss arugula, oil, and lemon juice; season. Build sandwiches with arugula, sausage patties, and eggs.