Begin by bringing a large pot of salted water to a rolling boil. Add the pasta and cook according to package instructions until al dente. Drain the pasta and set aside, reserving a small cup of pasta water in case you need to adjust the creaminess later.
In the same pot, heat the extra virgin olive oil over medium heat. Once hot, add the diced onion, carrots, and celery. Sauté the vegetables for about 5 minutes, or until they soften and the onion becomes translucent.
Add the minced garlic to the pot and sauté for an additional minute, stirring frequently until the garlic is fragrant but not browned.
Incorporate the shredded cooked chicken into the pot, stirring to combine with the sautéed vegetables. Gradually pour in the chicken broth, bringing the mixture to a gentle simmer.
Lower the heat and mix in the heavy cream, dried thyme, dried rosemary, and season with salt and freshly ground black pepper. Allow the mixture to simmer for 5-7 minutes, stirring occasionally, until it thickens slightly.
Fold in the cooked pasta along with the frozen peas, stirring well to ensure all components are thoroughly combined and warmed throughout. If the dish appears too thick, add a splash of the reserved pasta water to reach your desired consistency.
Remove the pot from heat and stir in the shredded sharp cheddar cheese, allowing it to melt into the creamy sauce, creating a luscious finish.
Serve the dish hot, garnished with chopped fresh parsley for a touch of color and a burst of freshness.