Preheat the oven to 350˚F (178˚C). Pick your desired baking dish that can fit the pasta tubes in one layer. Lightly grease the baking dish with cooking spray or vegetable oil. Set aside.
In a large pan over medium-high heat, add ½ tablespoon olive oil. Add the chicken cubes and cook until no longer pink and slightly golden brown. Season with salt and pepper. Add black garlic and sauté until fragrant. Add spinach and cook until just wilted. Taste and add more salt if needed. Transfer to a large bowl. Stir in the lemon juice and mozzarella cheese. Let it cool for 10 minutes.
Bring a large pot of water (salted) to a boil over high heat. Cook the pasta until tender but still firm to the bite, stirring occasionally. Drain and rinse the pasta with water until cool. Drain. Pat dry with paper towels.
Stuff the pasta tubes with chicken filling. Place the stuffed pasta in the prepared baking dish.
To make the sauce, in a small saucepan over medium heat, add 1 tablespoon butter and 1 tablespoon olive oil. Add the black garlic and sauté until fragrant. Add the heavy cream. Bring to a simmer and cook until slightly thickened, about 5 minutes, stirring frequently. Remove from heat. Stir in the Parmesan and lemon juice. Season with salt and pepper. Mix well. Pour the sauce over the stuffed pasta evenly. Sprinkle the top with mozzarella cheese and 2 tablespoons Parmesan cheese. Bake in the oven for 15 – 20 minutes or until the sauce is bubbling.