Pat each chicken breast dry with a paper towel. Season each all over with 1/2 teaspoon salt.
Line a rimmed baking sheet with aluminum foil, turning the ends up to keep in juices while cooking. Preheat the oven to broil and set an oven rack 6 inches from the heat.
Add the mushrooms to the prepared baking sheet. In a small, microwave-safe bowl, melt 4 tablespoons of the butter in the microwave. Drizzle it over the mushrooms, gently toss to coat, then spread the mushrooms in one layer. Nestle the chicken breasts, skin-side up, between the mushrooms, then broil the chicken and mushrooms 6 inches from the heat until the chicken skin is crisp and deep golden-brown but not burned, about 10 minutes.
Turn down the heat to 250°F. Continue cooking until the chicken is cooked through, about 25 minutes.
Meanwhile, heat 1 tablespoon of the remaining butter in a large skillet over medium heat, add the spinach. Cook, stirring often until the spinach starts to wilt. Season to taste with salt and pepper. Keep warm on the stovetop.
Halve each cooked chicken breast widthwise, then divide the breasts among 4 dinner plates.
Gently stir the remaining 2 tablespoons butter and the lemon juice into the mushrooms. Season to taste with salt and pepper. Spoon the mushrooms and sauce around the plated chicken, divide the spinach among the 4 plates, and serve.