Adjust an oven rack to the middle position and heat the oven to 200 degrees. Spread the flour in a shallow dish.
Pound the thicker ends of the breasts as needed to make chicken even. Pat dry with paper towels, then season with salt and pepper. Dredge both sides of the chicken in flour and shake off excess.
Heat 2 tablespoons of the oil in a large skillet over medium-high heat until just smoking. Add the chicken and cook until light golden brown on both sides, about 10 minutes. Transfer the chicken to a plate and keep warm in the oven.
Add the remaining 1 tablespoon oil to the skillet and return to medium-high heat until shimmering. Add the pancetta and mushrooms. Cook until the pancetta is crisp and the mushrooms are brown, about 10 minutes.
Stir in the garlic and tomato paste. Cook until the tomato paste begins to brown, about 1 minute. Stir in the Marsala, scraping up any browned bits, and simmer until reduced and slightly syrupy, about 8 minutes.
Stir in the lemon juice, and any accumulated chicken juice from pan in the oven along with the chicken. Turn the heat to low and whisk in the butter, one piece stirring until mixed. Add chicken to pan. Off the heat, stir in the parsley and season with salt and pepper to taste.