Preheat the oven to 375°F (190°C) for the sausages.
Place the sausage links on a baking sheet and bake until cooked through to an internal temperature of 160F / 71C.
In a small bowl, prepare a simple vinaigrette with olive oil, lemon juice, salt, and pepper. Toss the arugula with the vinaigrette.
Toast the bread slices until golden brown. Spread butter on each slice.
Melt butter in a non-stick skillet over medium heat. Crack the eggs into the skillet and cook to your desired doneness (sunny-side-up, over-easy, etc.). Season with salt and black pepper to taste.
Serve the eggs with arugula salad, sausage, toast and orange juice.