In a 12-inch skillet, heat 3 tablespoons olive oil over medium heat until just smoking. Add the onion, capers, and anchovy paste, and cook until the mixture is golden brown, about 8 to 10 minutes. Add the chicken livers and stir until lightly browned, about 4 to 5 minutes. Add the wine and balsamic vinegar. Bring the mixture to a boil. Lower the heat and let the mixture simmer for 15 minutes. Season, to taste, with salt and pepper. Remove from the heat.
Using a potato masher, mash the mixture so that it is lumpy. Stir in the chili flakes and parsley.
Toast the bread until golden brown and place on serving plates. Spread equal amounts of the liver mixture over 1 side of each piece of bread and serve immediately.