Bring a pot of water to a boil for "riggies". Salt water and cook pasta to al dente. Heads up: you will need to reserve cup of cooking water just before you drain the pasta.
Trim up the thighs and chop all the chicken into bite size pieces. Season the chicken with salt and pepper.
Heat 3 tablespoons extra-virgin olive oil in a large high sided skillet over medium-high to high heat.
Add chicken to hot pan and brown all over, 7 to 8 minutes then remove. Add onions, 3 to 4 cloves of garlic, cubanelle peppers then season with salt and pepper and cook 6 to 7 minutes to soften then add roasted red peppers. Cut tops off the hot peppers and scoop out the seeds with a teaspoon then chop the hot peppers and add them to the skillet. Stir in tomatoes and add chicken back to the sauce. Bring sauce to a bubble then stir in some starchy cooking water from the pasta - about a cup, then add the cream, twice around the pan, about 1/3 to 1/2 cup. Reduce heat to lowest setting and simmer a couple of minutes. Add pasta and 1/2 cup cheese and toss to combine.
While the chicken and peppers are cooking, heat 3 tablespoons extra-virgin olive oil in large skillet over medium-high heat. Add chopped ham and cook 2 minutes, add reserved pasta water and garlic and reduce heat a bit. Add greens and wilt down, 2 minutes. Season the greens with salt and pepper, add stock. Simmer greens 5 to 6 minutes and serve.
Top the pasta with shredded basil. Pass cheese to top both pasta and greens and serve with crusty bread.