Place the potatoes in a roasting pan. Drizzle with the oil. Season with 3/4 teaspoon of the salt and the pepper, and toss. Roast for 30 minutes, shaking the pan once. Add the asparagus to the pan with the potatoes, season with the remaining salt, and toss. Roast until the asparagus is tender, 12 to 15 minutes.
Meanwhile, in a saucepan, bring the broth and vinegar to a boil. Rinse the chicken and pat it dry with paper towels. Pound the chicken to an even thinness. Add the chicken and, if necessary, enough water (up to 1/2 cup) to cover it. Simmer for 1 minute. Cover, remove from heat, and set aside until cooked through, about 15 minutes. Transfer the chicken to a cutting board. Return the liquid to medium-high heat and simmer until reduced to about 1/3 cup, about 10 minutes. Thickly slice the chicken. Divide the ingredients among individual plates. Drizzle with the balsamic mixture.
Tip: The pungent smell of simmering vinegar can linger long after you remove the pan from the stove. To reduce the odor, open the windows or turn on a fan early in the cooking process.