Using the flat-side of a meat mallet or bottom of a heavy saucepan, pound the chicken breasts to ½ inch thick. Season with ¼ teaspoon kosher salt and ¼ teaspoon freshly ground black pepper. Lightly coat the chicken breasts with flour. Heat 1 tablespoon olive oil in a large skillet on medium and cook chicken until golden brown, 4 to 5 minutes per side; transfer to a plate.
While the chicken is cooking, finely chop the shallot and garlic.
Add 1 tablespoon olive oil to same skillet. Cook the sliced cremini mushrooms on medium-high, tossing occasionally, until golden brown, 5 minutes. Add the chopped shallot and garlic. Season with ¼ teaspoon salt and ¼ teaspoon freshly ground black pepper; cook 2 minutes.
Add the low-sodium chicken broth and Marsala wine to the skillet along with the browned chicken and its juices and simmer until the liquid is reduced by half, 4 minutes. Sprinkle with chopped parsley. Serve with sautéed spinach.
