Bring a large pot of salted water to a boil and cook 12 ounces penne pasta according to package directions until al dente; drain and set aside.
Heat 1 tablespoon olive oil in a large heavy-bottomed pot over medium-high heat.
Add 1/4 cup chopped green onions and the sliced red, yellow, and green bell peppers; sauté until the peppers are tender and beginning to brown, about 5–7 minutes.
Stir in 2 cups cooked shredded chicken, 3 minced garlic cloves, and 3 tablespoons jerk seasoning; season with salt and pepper to taste and cook, stirring, for about 3 minutes to combine flavors.
Pour in 1 cup chicken stock and 1 cup heavy cream, then add the cooked pasta. Bring to a boil while stirring, then reduce heat to medium and simmer until the sauce thickens slightly and the pasta cooks a little more, about 5–8 minutes.
Remove from heat and serve immediately, garnishing with additional sliced green onions if desired.