Heat oil in a large Dutch oven over medium.
Sprinkle chicken evenly with 1 teaspoon of the Cajun seasoning.
Add to hot oil; cook until browned on both sides and cooked through, about 8 minutes per side.
Remove from Dutch oven; let rest 5 minutes. Cut into ¾-inch cubes.
Whisk together flour and ¼ cup of the half-and-half in a small bowl; set aside.
Add garlic to Dutch oven; cook over medium, stirring often, for 1 minute.
Add broth and remaining 1 ¼ cups half-and-half; cover and bring to a boil over medium-high.
Break linguine noodles in half; add to Dutch oven.
Reduce heat to medium; cover and cook, stirring occasionally, 8 minutes.
Stir in chicken cubes and reserved flour mixture; cook, tossing constantly, until pasta is well coated and heated through, about 2 minutes.
Remove from heat; stir in cheese and remaining 1 teaspoon Cajun seasoning until cheese is melted.
Sprinkle with parsley; serve.
