Coat the chicken in 1 tablespoon balsamic vinegar and 1 tablespoon EVOO and season with salt and pepper. In a large skillet, heat 1 tablespoon EVOO, 1 turn of the pan, over medium-high heat. Add the chicken and cook for 5 minutes, then lower the heat to medium, flip the chicken and cook for 7 minutes more. Transfer to a cutting board to rest for a couple of minutes.
While the chicken cooks, heat another large skillet over medium heat and add the remaining 2 tablespoons EVOO, 2 turns of the pan. Add the mushrooms and cook, stirring often, until dark brown, 10 to 12 minutes. Stir in 1 tablespoon rosemary; season with salt and pepper. Scoot the mushrooms off to the side of the pan and add the butter to melt. Whisk in the flour and cook for 1 minute. Whisk in the remaining 1 tablespoon vinegar, then the chicken broth. Stir the mushrooms back in and simmer until the gravy is thickened, 2 minutes. Add the remaining 1 tablespoon rosemary and salt and pepper to taste.
Toast the bread and cut in half on an angle; arrange on 4 plates. Dress the arugula with a squeeze of lemon juice, a drizzle of EVOO and a sprinkle of salt and pepper. Slice the chicken on an angle. Top the toasts with the mushroom gravy, chicken and arugula.