In a small shallow bowl, whisk together the flour, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Set aside.
If needed, pound the chicken breasts to an even thickness. Pat chicken dry with paper towel and season on both sides with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Dredge the breasts in the flour mixture (dip each breast in the flour to coat on both sides). Shake off any excess flour. Set aside.
In a large skillet over medium-high heat, heat 1 tablespoon butter and 1 tablespoon oil. When butter is melted and foaming subsides, place floured breasts in pan in one layer and sauté until golden brown, around 2-3 minutes per side. Transfer to a plate; do not clean the skillet.
In the same skillet over medium heat, heat 2 tablespoons butter and 1 tablespoon oil. When butter is melted and foaming subsides, add shallot and sauté until soft, around 2 minutes.
Add garlic and cook for 30 seconds more, then add mushrooms and cook without stirring too much, until golden brown, 5-7 minutes. Season with 1/8 teaspoon salt after they’ve started turning brown (around 3-4 minutes).
Stir in white wine, bring to a boil then immediately reduce to a simmer for 1-2 minutes, stirring and scraping any browned bits off the bottom of the pan. Stir in chicken broth and simmer for an additional 2-3 minutes.
Stir in 2 tablespoons lemon juice, soy sauce, and capers. Nestle the chicken breasts in the sauce and add any chicken juices that have accumulated on the plate. Simmer gently for approximately 5-7 minutes or until the chicken reaches 165 degrees F on an instant-read thermometer inserted into the middle of the thickest portion of the breast.
Transfer chicken to a plate and tent with foil to keep warm. Remove sauce from heat and stir in 1 tablespoon butter. Taste and adjust seasonings as needed (additional lemon juice, more salt or pepper, more soy sauce).
To serve, place chicken atop cooked buttered pasta noodles or mashed potatoes, either on individual serving plates or a serving platter. Spoon sauce on top of chicken. Garnish with parsley.