Place black-eyed peas in a large pot and cover with 6 cups water. Bring to a boil. Add lemon slices and sugar; simmer until black-eyed peas start to soften, about 30 minutes. Drain and rinse.
Return black-eyed peas to the pot. Cover with 1 inch water. Bring to a boil. Simmer until almost tender, 30 to 40 minutes. Add zucchini, carrots, and salt. Simmer until black-eyed peas and carrots are soft, 20 to 50 minutes.
Garnish black-eyed peas with tomatoes, onion, olive oil, lemon juice, and capers.