In a 12" skillet over medium heat, heat oil. Cook sausage, turning occasionally, until golden brown, 3 to 4 minutes per side. Transfer to a plate.
Meanwhile, in a large pot of boiling salted water, cook pasta, stirring occasionally, until very al dente, about 2 minutes less than package directions. Reserve about ½ cup pasta water before draining.
In same skillet over medium-high heat, combine 1 tablespoon butter and a splash of broth. Cook, scraping browned bits from bottom of pan, until butter is melted. Add garlic and tomato paste and cook, stirring, until fragrant, about 1 minute. Add Cajun seasoning and remaining 1 ½ cups broth. Cook, stirring, until combined, about 1 minute. Slowly whisk in cream and bring to a boil. Reduce heat to medium-low and bring to a simmer. Cook, whisking constantly and adding 2 cups Parmesan in a slow, steady stream, until sauce is slightly thickened and reduced by about one-third, about 5 minutes. Add remaining 1 tablespoon butter and whisk until melted.
Add pasta to skillet and toss, adding tablespoons of pasta water as needed and up to ½ cup more Parmesan, until pasta is covered in sauce and sauce is glossy and smooth.
Return sausage to skillet. Top with parsley and Parmesan.
