Sprinkle 2 tablespoons cheese over bread and heat in toaster oven until cheese melts, about 3 minutes. Divide among 4 plates.
Meanwhile, heat 2 tablespoons olive oil in a large, heavy-bottomed skillet over medium-high heat. Add anchovies, mashing with the back of a spoon to form a paste. Add chard, season with a healthy pinch of salt and pepper, and cook, tossing with tongs, until stem ends are tender and leaves are bright green and soft, about 10 minutes. Add 1/2 cup water if greens are sticking to pan. Remove greens from pan and divide among plates beside toast.
Add 1 tablespoon olive oil to skillet, set over medium-high heat, and crack eggs directly into pan. Cover and cook until yolk is set to desired consistency, 2 to 5 minutes.
Serve eggs over toast and greens, and top with capers and remaining cheese. Add a squeeze of lemon juice, remaining olive oil, and salt and pepper to taste.