Place each chicken breast half between 2 sheets of heavy-duty plastic wrap; pound to 1/2-inch thickness using a meat mallet or rolling pin. Sprinkle chicken evenly with 1/4 teaspoon salt.
Heat oil in a large nonstick skillet over medium heat. Add chicken; cook 6 minutes on each side or until done. Remove chicken from pan; keep warm.
Add broth, chili sauce, and fish sauce; bring to a boil, scraping pan to loosen browned bits. Cook until broth mixture is reduced to 2/3 cup (about 4 minutes). Remove from heat; add lime juice and peanut butter, stirring until smooth. Serve sauce over chicken; sprinkle with peanuts. Garnish with lime wedges, if desired.