Be sure to remove excess fatty skin from the thick ends of the breasts when trimming. Harissa, a chile paste, can be found in the international aisle of many supermarkets, usually with Middle Eastern or Indian ingredients.
Total Time: 1¼ hours
1. Adjust oven rack to lower-middle position and heat oven to 325 degrees. Working with 1 breast at a time, use your fingers to carefully separate skin from meat. Peel back skin, leaving skin attached at top and bottom of breast and at ribs. Sprinkle ¾ teaspoon salt evenly over chicken (⅜ teaspoon per breast). Lay skin back in place. Using metal skewer or tip of paring knife, poke 6 to 8 holes in fat deposits in skin of each breast. Spray skin with oil spray.
2. Place chicken, skin side down, in 12-inch ovensafe skillet and set over medium-high heat. Cook, moving chicken as infrequently as possible, until skin is well browned, 7 to 9 minutes.
3. Carefully flip chicken and transfer skillet to oven. Roast until chicken registers 160 degrees, 25 to 30 minutes.
4. Transfer chicken to plate; do not discard liquid in skillet. Add shallot, harissa, and remaining ½ teaspoon salt to skillet and cook over medium-high heat, stirring occasionally and scraping up any browned bits, until moisture has evaporated and mixture begins to sizzle, 1 to 3 minutes. Add carrots and water and bring to simmer. Cover skillet, reduce heat to medium, and cook until carrots are tender, 10 to 12 minutes, stirring halfway through cooking. Uncover and continue to cook, stirring frequently, until sauce begins to coat carrots, 2 to 4 minutes longer. Add lemon juice, 1 teaspoon mint, and any accumulated chicken juices to skillet and toss to combine. Season with salt to taste. Transfer carrots to serving platter and sprinkle with remaining 1 teaspoon mint. Top with chicken and serve.