Wash and strain the black-eyed peas. Wash and strain the fennel. If using fennel bulb, trim it and chop it as for the onion. If using spinach and/or chard, trim, coarsely chop and wash.
Place the black-eyed peas in a large pot with enough water to cover by about two inches. Bring to a boil. Lower heat and simmer.
In the meantime, heat 1/3 cup olive oil in a deep, heavy skillet or pot, and saute the onion and fennel over low heat until wilted and tender, about 10-12 minutes. Add the greens if using, and give them a few stirs until wilted but al dente. You want to bring out some of their flavor.
Remove and add to the black-eyed peas, stirring gently to combine. Stir in the tomatoes and remaining olive oil. Simmer, slightly covered, for about 45-50 minutes, or until the black-eyed peas are tender. Add more water if necessary during cooking. About ten minutes before removing from heat, season with salt and pepper. Squeeze in a little lemon juice or add a touch of vinegar, to bring up the flavors a bit.
Remove, cool to room temperature, and serve.