Heat 3 teaspoons of oil in a large nonstick pan over medium high heat; sauté the onions until golden brown (do not burn), about 20 minutes; stir in 1/4 teaspoon of salt.
In a small pot, whisk together the broth, vinegar, and mustard and heat over medium low heat. Add the cooked onions to the broth with 1/8 teaspoon of the pepper and cook until liquid is reduced to half, about 4-5 minutes. Remove pot from heat and keep warm.
Thin chicken breasts to 1/4 inch thickness by placing them between 2 sheets of heavy duty plastic wrap and pounding with a rolling pin or meat mallet. Sprinkle each with remaining 1/4 teaspoon of salt and 1/8 teaspoon of pepper.
Place the thyme in a shallow dish or plate and dredge chicken in the thyme, press to secure leaves.
Heat the remaining teaspoon of oil over medium high heat in the same large pan from step 1. Add in the chicken and cook until no longer pink, about 2-3 minutes per side.