Quarter eggplant lengthwise then slice 1/4- to 1/2-inch thick. Salt and drain in colander 30 minutes; press out excess liquid.
Broil the peppers to char the skins and let cool in a covered bowl. Peel and seed the peppers, and coarsely chop. If you have a gas stove, peppers can be charred over open flame.
Bring a large pot of water to a boil for the pasta.
Heat a large pot over medium to medium-high heat with olive oil, 4 turns of the pan. When the oil is hot, add the anchovies and stir until they melt into the oil. Add garlic, chili pepper or flakes, eggplant, stir to combine and partially cover the pan to sweat it for a few minutes until light brown at the edges. If using fresh tomatoes, add them to the pan and cover pan tightly; shake every few minutes until tomatoes burst. If using canned tomatoes, hand crush and add to eggplant along with the juice. Stir in basil, parsley, chopped peppers and capers, and season with salt and pepper to taste.
Salt pasta water and cook pasta to al dente. Add a cup of starchy water to the pasta sauce just before draining. Toss pasta with sauce and add additional cooking water, 1/4 cup at a time, until pasta is coated evenly.
Serve in shallow bowls immediately or top with a little cheese, if you like. (We usually skip cheese for dishes that include anchovies but I am a sucker for ricotta salata on this one.