Fill a large pot with water, season generously with salt, and bring to a boil. Cook the pasta until just tender but still firm to the bite, then drain.
While the pasta cooks, remove the skin from the rotisserie chicken meat, then shred the meat into bite-sized pieces. Season the shredded chicken with 1½ teaspoons pepper and salt to your taste. Thinly slice the chicken skin, chop it finely, and set aside.
Heat a Dutch oven or a large heavy-bottomed pot over medium heat and melt the butter. Add the sliced onions, season with salt and pepper, and cook while stirring frequently and pressing them down with a spoon until they become very soft, translucent, and develop golden edges, about 10 minutes. Add the chopped garlic and Dijon mustard, and continue cooking until fragrant and softened, about 2 minutes more.
Increase the heat to high, pour in ¾ cup of the chicken stock and the heavy cream if using, scraping the pot's bottom to loosen any browned bits. Let it simmer gently, then add the drained pasta, seasoned chicken, and chopped chicken skin. Incorporate the greens gradually in small handfuls, stirring until they wilt and soften. Remove from heat, stir in the lemon zest, and squeeze the juice from the lemon halves over the pasta. Add the remaining ¼ cup of stock if you prefer a saucier dish.
Top with freshly grated Parmesan cheese and serve immediately, adding extra salt and pepper as desired.