Slice the chicken breasts in half horizontally so that you have 8 thinner chicken breast cutlets. Sprinkle both sides of each cutlet with salt and pepper.
Heat 2 tablespoons butter and 1 tablespoon olive oil in a large skillet over medium-high heat until hot. Add half the chicken cutlets and cook until golden brown and completely cooked through, 3 to 4 minutes per side. Remove the cutlets to a plate and set aside. Cook the remaining cutlets in the same manner.
Reduce the heat to medium. Add the garlic to the skillet and stir until golden brown, about 1 minute. Carefully pour in the wine and allow it to simmer and reduce by half, 1 to 2 more minutes.
Stir in the mustard, followed by the chicken broth. Add the remaining 2 tablespoons butter and stir until the sauce begins to thicken, 1 to 2 minutes. Season the pan sauce with salt and pepper. Add the herbs and stir.
Return the chicken cutlets to the skillet, turning them in the sauce to coat and to rewarm them. Serve on top of the Garlic Spinach. Spoon more sauce from the skillet over them and garnish each plate with more fresh herbs.
Heat the olive oil in a large, high-sided skillet over medium-high heat until hot, then add the garlic and give it a quick stir. Add the spinach and let it cook down, turning often to bring the wilted pieces to the top, until softened, a minute or two.
Cook until completely wilted, a minute or two more, then season the spinach with salt and pepper, as well as the lemon juice. Toss to evenly combine everything.