Place the potatoes and 1 tablespoon of the salt in a large pot. Add enough cold water to cover by 2 inches. Bring to a boil, reduce heat to a simmer, and cook until tender, about 20 minutes. Drain and set aside until cool enough to handle.
Heat oven to 400° F. Rinse the chicken and pat it dry with paper towels, then place it in a baking dish lined with foil. Whisk together the mayonnaise and 2 tablespoons of the pesto in a small bowl. Spread the mixture evenly over the top of each chicken breast. Bake until the chicken is cooked through, about 10 minutes. Leave the chicken in oven and turn on broiler. Broil until golden brown, about 3 minutes. Meanwhile, peel and cut the potatoes into large chunks. Toss them with the pepper and the remaining pesto and salt. Sprinkle the chicken with the pine nuts and raisins and serve with the potatoes.
Tip: To toast the pine nuts, place them in a dry skillet over medium heat for about 3 minutes, or on a baking sheet in a 350° F oven for about 7 minutes, stirring once.