Coat a large nonstick skillet with cooking spray. Add shallot, 1 tablespoon ginger, and 1 teaspoon garlic; sauté 3 minutes or until shallot is tender. Add curry powder and 1/4 teaspoon crushed red pepper; sauté 1 minute. Stir in broth, peanut butter, 2 tablespoons lime juice, 1 tablespoon soy sauce, brown sugar, and 1/4 teaspoon salt. Simmer 3 to 5 minutes, or until mixture thickens, stirring constantly. Transfer mixture to a bowl; cover and chill at least 2 hours.
Cut each chicken breast half lengthwise into 4 strips. Place strips between 2 pieces of heavy-duty plastic wrap, and flatten to 1/4-inch thickness, using a meat mallet or rolling pin. Combine remaining ginger, garlic, crushed red pepper, lime juice, soy sauce, salt, and sesame oil in a heavy-duty zip-top plastic bag. Add chicken, and seal bag securely. Marinate in refrigerator 1 1/2 hours, turning once.
Soak wooden skewers in water 30 minutes; drain.
Remove chicken from marinade, discarding marinade. Thread chicken onto wooden skewers. Place skewers on a baking sheet coated with cooking spray. Broil 2 to 3 minutes on each side or until done. Serve with peanut dip.