1. Grate potatoes into large bowl filled with water. Swish potatoes around in water. When water is cloudy, drain and turn potatoes into a towel; wring out moisture.
2. Combine grated potatoes, anchovy, onion, salt and pepper in large bowl.
3. Heat 2 tablespoons oil in large cast-iron skillet over medium-high heat. Add the potatoes; pack down with back of the spatula; cook until browned. Turn to brown other side. Add more oil if necessary to prevent sticking. Keep chopping with the spatula and turning potatoes until browned throughout, about 20 minutes total. Add roasted peppers, garlic and basil; cook another minute. Serve immediately.