Bring a large pot of water to a boil. Add pasta and cook al dente according to package directions. Drain the pasta and set it aside.
In a blender, combine vegetable broth and cashews. Blend until smooth, creating a creamy cashew sauce. Pour sauce through a fine-mesh strainer into a small bowl.
Heat olive oil in a large high-sided skillet over medium-low heat. Add mushrooms, garlic, and rosemary. Cook, stirring frequently, until the mushrooms are tender and the garlic is fragrant, about 2 to 3 minutes.
Add cashew sauce, milk, and soy sauce to skillet. Bring to a boil over medium heat, then reduce the heat to medium-low and simmer, stirring frequently, until sauce has thickened, 4 to 5 minutes.
Stir in shredded chicken and continue to simmer, stirring occasionally, for an additional 2 to 3 minutes.
Add cooked pasta to the skillet, tossing gently in the creamy sauce until completely coated.
Top the Creamy Mushroom & Rotisserie Chicken Pasta with grated Parmesan cheese and a sprinkle of crushed red pepper for a touch of heat.