Pat the chicken dry with paper towels and season with salt and pepper.
In a wide pot over medium-high heat, warm the olive oil. Working in batches if necessary to avoid crowding, add the chicken pieces and brown, turning as needed, until golden brown on all sides, 10 to 12 minutes. Transfer chicken to a plate.
Pour off all but 2 tablespoons of the fat from the pot, reduce the heat to medium, and add the onion, carrots, celery and about 1/2 teaspoon salt. Stir well, scraping the bottom of the pan to loosen the brown bits, and cook, stirring occasionally, until onion is translucent and soft and carrots and celery are bright in color and beginning to get tender, 7 to 10 minutes. Add the garlic and 1 tablespoon of the chopped mint. Stir and cook 1 minute more, being careful not to let the garlic brown. Remove from heat.
Using a microplane grater, and working over the pot, grate the zest from 2 of the lemons, capturing both the zest and the spray of citrus oils in the pot. Juice lemons, enough to fill 3/4 cup. Add 1/2 cup of the lemon juice, with the tomatoes and 1 cup water (or chicken broth). Return pot to medium heat and cook until liquid is reduced by one-third, about 5 minutes. Return chicken to pot and spoon some of the liquid over it. Cover pot, reduce heat to low, and simmer, basting the chicken occasionally with the liquid (the chicken should always be half submerged in the sauce as it cooks), until chicken is very tender, about 45 minutes. Sauce develops as the dish cooks, so you may not need additional liquid, but add a little water or broth if needed.
Uncover the pot during the last 10 minutes of cooking to thicken the liquid. At this point, taste the sauce. It should be very lemony. Add more lemon juice as needed, along with the remaining 3 tablespoons mint. If the sauce seems too thin, add the potato and cook until the potato dissolves and the sauce is thick, about 10 minutes. Serve in shallow bowls with rice or another grain.