Place 2 tablespoons water and potatoes in a microwave-safe bowl; cover and microwave at HIGH 4 minutes or until almost tender.
Heat 2 tablespoons oil in a large nonstick skillet over medium-high. Sprinkle chicken with 1/2 teaspoon salt and 1/2 teaspoon black pepper. Add chicken to pan; cook 5 minutes on each side. Remove from pan; keep warm.
Cut half of chile into thin slices; finely chop remaining half. Add potatoes and chile slices to drippings in pan; cook 4 minutes. Stir in 1/4 teaspoon salt and 1/8 teaspoon black pepper.
Place chopped chile, parsley, cilantro, shallots, and garlic in a mini food processor; pulse to combine. Add remaining 3 tablespoons oil, 1/4 teaspoon salt, 1/8 teaspoon black pepper, and juice; process until smooth. Serve with chicken and potatoes.