In large bowl, combine cod with water to cover by 2 inches. Cover and refrigerate, changing water 3 times a day, 1 to 3 days (see cooks' note, below). Refrigerate until ready to use.
In small bowl, combine capers with water to cover by 1 inch. Cover and refrigerate 1 day, changing water 3 times.
Drain and rinse capers, then squeeze dry. Drain cod and pat dry, then cut into 4- by 2-inch pieces.
In heavy 6- to 8-quart pot over moderate heat, heat 3 tablespoons oil until hot but not smoking. Add onion and sauté until softened and golden, 3 to 4 minutes. Add red pepper flakes, capers, tomatoes and juice, and salt. Using wooden spoon, gently crush tomatoes. Bring to simmer, then carefully add cod. Add parsley and 1/4 cup mint and gently spoon some of sauce over cod. Simmer, uncovered, until cod is tender and heated through, about 5 minutes.
Using slotted spatula, transfer cod to serving platter. Remove pot from heat and stir in 3 tablespoons olive oil. Spoon sauce over fish, then sprinkle with remaining 1 tablespoon mint and 1 tablespoon olive oil. Serve warm or at room temperature.
Chef's note: Brands of cod differ in their degree of saltiness: A less salty one may need only 1 day of soaking, while another could require up to 3. To test it, simply taste a small piece after 1 day; you want it to be pleasantly salty but not overwhelming.