Preheat broiler with rack in highest position. In a shallow dish, whisk together 2 tablespoons olive oil and vinegar. Season with salt and pepper. Dip sandwich bread in vinaigrette and arrange in a buttered baking dish. Use a glass or 2-inch round cutter to remove centers of bread. Broil toast until beginning to brown on one side, about 2 minutes. Flip toast slices, crack eggs into holes, and return dish to broiler. Broil toast with eggs 2 minutes, then cover dish with foil. Broil eggs until just set, 3 to 4 minutes more.
While toast and eggs cook, heat remaining 1 tablespoon oil in a large skillet over medium-high. Add arugula and cook until just wilted, about 2 minutes. Divide arugula among plates and top with toast slices.
Before her diagnosis, Caroline wrote a book on cakes called Cake Magic!. She started developing a birthday cake using her gluten-free mix found in that book. Check out other recipes she’s developing for her new life—and the stories behind them—on her blog, The Wright Recipes. Her next book, Catalan Food, written with chef Daniel Olivella, comes out in early September from Clarkson Potter.