Add the oil to a dutch oven, or other large pot. Heat over medium heat for a minute or two, when the oil is hot add the chicken and the Cajun seasoning. Cook for 5 minutes, stirring occasionally, then add in the garlic and stirring often, so it doesn't burn and turn bitter, continue cooking for another minute.

Stir in the milk and broth, scraping up and browned bits or spices stuck to the bottom of the pan. Add the pasta to the pot, stirring as you do so it doesn't clump and stick together. Cover and bring to a rapid, rolling boil, continuing to stir the mixture occasionally.

Reduce the heat to medium-low and let the pasta simmer, covered, for 9-11 minutes, stirring every 2-3 minutes to prevent sticking. Cook until the time's up, or until there's only 1/2 inch of liquid left in the bottom of the pot.

Remove the pot from heat and stir in the cheese until melted and evenly incorporated. Season with salt & pepper, and serve immediately.