In a small blender, combine milk, grated cheese, and cream cheese and set aside.
Season chicken with cajun seasoning.
Heat a large pot over high heat. When the pan is very hot, spray it with oil and add the chicken, season with salt and pepper to taste, and cook, spreading it in a single layer on high heat until the bottom browns, about 3 minutes, then flip to cook the other side until the chicken is browned on all sides and cooked through, about 2 to 3 minutes. Transfer to a plate.
Add a splash of the broth and scrape any browned bits from the bottom of the pot. Add the remaining broth, cream cheese mixture, the pasta, and leeks. Stir to combine and bring to a boil, then cover, reduce the heat to medium-low, and simmer, stirring occasionally to help the pasta cook evenly, for 15 minutes.
Taste the pasta and if it’s not al dente, or there is still a lot of liquid at the bottom of the pot, cover and cook for another minute and check again.
When the pasta is ready, add the asparagus and artichokes, gently stir, and cover. Cook until the asparagus is crisp-tender, the pasta is tender, and the liquid is mostly absorbed, about 3 minutes.
Uncover and add the chicken and any juices that have accumulated and the peas and stir to combine. Cook just for about 30 seconds to warm the peas through.
Remove from the heat and stir in the basil. Serve with more parmesan cheese, for passing.
