Mix together the butter, basil, parsley and garlic in a small bowl. Sprinkle with 1/2 teaspoon salt and 1/4 teaspoon pepper.
Heat a 9- or 10-inch cast-iron skillet over medium-high heat. Melt 2 tablespoons of the compound butter in the skillet. Add the mushrooms and cook, stirring occasionally, until the mushrooms are browned and have released their liquid, 3 to 4 minutes. Transfer the mushrooms to a bowl and set aside.
Melt 2 more tablespoons of the compound butter in the skillet. Remove from the heat and cover the bottom of the skillet with baguette rounds. Place in the oven and bake until the bottoms are browned and toasted, 15 to 20 minutes.
Meanwhile, place the chicken breasts flat on a work surface. Sprinkle both sides lightly with salt and pepper. Mix together the mozzarella, mayonnaise and 1/4 cup of the Parmesan in a medium bowl. Spread a heaping tablespoon of the mixture on a chicken breast with a spatula or spoon. Top the mixture with 6 to 7 spinach leaves and a few mushrooms. Roll up the chicken, starting with the narrowest side to create a short, thick roll. Cut the roll in half crosswise. Repeat with the remaining chicken breasts and filling.
Remove the skillet from the oven. Flip the baguette rounds. Arrange the chicken cut-side up on top of the toasted baguette rounds. Add a few pats of compound butter on top of the chicken. Sprinkle the rounds with the remaining 1/4 cup Parmesan and bake until the chicken is fully cooked, 40 to 45 minutes.
Top the asparagus with any remaining compound butter and serve with the roll-ups.