Making these spirals involves nothing more complicated than topping pounded chicken breasts with pesto, rolling them up, and securing them with toothpicks.
Toast almonds in a skillet over medium heat, tossing frequently, until golden and fragrant, about 6 minutes. Let cool completely. Process almonds and shallot in a food processor until a coarse paste forms. Add 1 cup mint; pulse a few times until coarsely chopped. Add cheese and oil; pulse a few times until combined. Transfer to a small bowl.
One at a time, place chicken breast halves on a cutting board between two sheets of plastic wrap. Pound with the flat side of a meat mallet or the bottom of a small, heavy skillet until 1/8 to 1/4 inch thick. Sprinkle each piece all over with 1/8 teaspoon salt; season with pepper.
Put breast halves on a cutting board, smooth side down, and spread evenly with pesto. Roll up chicken, starting with the pointed tip of each piece. Secure with two toothpicks, pushing them in sideways at an angle; set aside.
Coat a 12-inch nonstick skillet with cooking spray, and heat over medium-high heat until hot but not smoking. Add zucchini and garlic; season with 1/8 teaspoon salt. Cook, stirring occasionally, until zucchini just turns golden brown, about 5 minutes. Reduce heat to medium, and cook, tossing occasionally, 5 minutes. Stir in vinegar; season with pepper. Transfer zucchini to a medium bowl, and cover with foil.
Wipe skillet clean. Coat with cooking spray; heat over medium heat until hot but not smoking. Add rolled chicken, and cook, turning to brown all sides, about 5 minutes. Reduce heat to medium-low, and continue cooking until chicken is no longer pink in the center, 20 to 25 minutes. Let rest 5 minutes, and cut each roll into 5 pieces. Toss reserved zucchini with 1/4 cup mint. Divide zucchini and chicken among four plates.