Combine green onions, garlic, vegetable oil, brown sugar, soy sauce, lemon zest, pepper flakes, salt, basil, lime juice, allspice berries, peppercorns, mustard, thyme, ginger, coriander, rosemary, and nutmeg in the bowl of a food processor; blend until a paste forms, scraping down the sides as necessary. Transfer jerk marinade to a large resealable plastic bag. Add chicken and mix well. Seal bag, and let stand at room temperature for 30 minutes, turning chicken in the bag periodically.

Bring a large pot of lightly salted water to a boil. Cook penne in the boiling water, stirring occasionally until tender yet firm to the bite, about 12 minutes. Drain.

Melt butter in a saucepan over low heat. Add milk and bring to a simmer. Cook for 5 minutes. Stir in Parmesan cheese and garlic, whisking constantly until cheese is completely melted and sauce is thoroughly combined. Set aside.

Cook chicken in a large nonstick skillet over medium-high heat and until it is no longer pink in the center and the juices run clear, 7 to 8 minutes. Transfer chicken to the saucepan with the Alfredo sauce. Bring to a simmer, and cook for 3 to 5 minutes, stirring constantly to keep the sauce smooth and to incorporate the jerk seasoning into the Alfredo sauce. Serve over pasta.