1. Spread mustard over one side of chicken breasts. Heat oil in a large nonstick skillet over medium-high heat until hot. Add chicken breasts, with mustard side down; cook 4 minutes. Turn chicken over. Reduce heat to medium.
2. Combine salsa and lime juice; spoon over and around chicken. Simmer uncovered until chicken is cooked through and sauce thickens, 6-8 minutes.
3. Transfer chicken to serving plates. Cook the remaining pan juices in skillet over high heat until slightly reduced, about 30 seconds.
4. Spoon juices over the chicken; top with avocado. Garnish with green onions, if desired.