In a large stockpot, boil 4 quarts water with salt and cook pasta until almost done, about 6 minutes. Before straining, reserve 1 cup of pasta water. Strain and set aside.
In a large skillet, add 1 tablespoon oil and heat over medium-high heat. Sear chicken in two batches and season each with ½ tablespoon jerk seasoning, browning quickly on the outside but not cooking all the way through about 4 minutes per batch. Remove the chicken from skillet and place on plate.
Heat remaining 1 tablespoon vegetable oil and add onions and peppers. Season with remaining 1 tablespoon of seasoning. Keep stirring vegetables as they continue to cook and don’t burn (reduce heat a bit if they start to cook too quickly). Cook until peppers and onions are soft and translucent with a little bit of caramelization and most of the moisture is cooked out, about 8 minutes. Add in garlic and thyme and cook for an additional minute.
Add in chopped tomatoes and scrape the bottom of the pan, removing all the browning from the bottom of the pan into the vegetables. Cook until all the moisture is gone, about 3 minutes.
Pour in chicken stock and return the chicken to the pan. Cover and cook until chicken is fully cooked, about 5-6 minutes.
Remove lid and add half and half. Stir together. Reduce heat to medium low and continue cooking until the sauce reduces and thickens a bit, coating everything, about 5-6 minutes. (If mixture becomes stodgy and the sauce has reduced too much, add a little pasta water to thin in out.) Stir in pasta and serve immediately.