Cook the pasta in a large pot of salted water according to package directions for al dente. Drain and set aside.
Mix 4 Tbsp. softened butter with the jerk seasoning and fresh thyme in a small bowl.
Melt the butter over medium heat in a large nonstick pan. Brown the chicken in the seasoned butter until golden brown on both sides, about 8-9 minutes per side. Remove chicken onto a cutting board. Slice into strips.
Add the mushrooms and asparagus to the pan and saute until tender. Remove onto a separate plate.
Lower the heat. Add the remaining butter to the pan. Melt the butter, then stir in the cream, Parmesan cheese and salt.
Add cooked chicken, vegetables and pasta back into the pan. Gently toss to coat everything in the sauce.