Preheat the oven to 400°F. Spray a rimmed baking sheet with nonstick cooking spray (you can line it with foil or parchment paper first for easier clean up).
Heat 2 tablespoons of the olive oil in a medium-sized skillet over medium heat, then add the mushrooms and sauté for about 10 minutes until they have given up any liquid, and that liquid has evaporated, and the mushrooms are tender and beginning to become crispy. Season with salt and pepper. Stir in the garlic and rosemary or thyme until fragrant, about 1 minute. Stir in the cream cheese and Parmesan until melted and well combined, and cool so that it’s just slightly warm or room temperature. Adjust the seasonings.
Loosen the skin from each thigh, and place a couple tablespoons of the mushroom mixture under the skin, spreading it out beneath the skin. Brush the skin with the remaining tablespoon of olive oil, and season with salt and pepper.
Roast for about 45 to 55 minutes, or until the chicken is cooked through (it should have an internal temperature of 165°F). Serve hot or warm.