Cook the pasta al dente according to the package instructions (it should have a bite to it). Reserve about 1/4 cup of pasta cooking water.
Prepare all the ingredients: Cut chicken breasts into 1-inch (2.5 cm) thick strips, cut bell peppers into thin strips, grate the Parmesan on the small holes of a box grater. Cut the white parts of green onions into thin slices and the green part into large chunks.
Rub the chicken with the Jerk seasoning, salt, and pepper.
Heat the oil in a large frying pan over medium-high heat. Add the chicken and cook for 1-2 minutes without stirring, letting the chicken brown, then cook, stirring, until the chicken is cooked through and browned. Transfer the chicken to a plate.
Add the bell peppers and green onions. Cook over high heat just briefly, until bell peppers are browned (don't cook them until soft).
Add the heavy cream, pasta cooking water, and chicken broth. Cook for a couple of minutes over medium heat until the sauce is thickened and bell peppers soft.
Add the chicken back to the pan, add grated parmesan, and cooked pasta. Stir everything together until well combined. Season with salt, pepper, and lemon juice.