In a large Dutch oven or cast iron braiser, melt 1 tablespoon of butter over medium-high heat. Add the shrimp and Cajun seasoning; cook, stirring occasionally, until the shrimp are pink and opaque, about 4-5 minutes. Use a slotted spoon to remove the shrimp to a plate.
Add the remaining 1 tablespoon of butter to the Dutch oven. When the butter melts, add the onion, bell pepper, and celery. Cook until the onion is translucent, about 5 minutes. Add the garlic; cook for 1 more minute. Stir in the milk and broth; bring to a simmer. Add the uncooked pasta, stirring frequently for about 3 minutes. Continue cooking until the pasta is al dente, about 8 more minutes for fettuccine or 6 more minutes for linguine, stirring occasionally.
Stir in the heavy cream, Parmesan, and green onions. Simmer until the sauce thickens and the cheese melts. Remove from the heat and stir in the cooked shrimp. Garnish with fresh parsley and serve.