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In a large bowl, toss shrimp, Cajun seasoning, and 2 Tbsp. oil; lightly season with salt.
In a large, deep, 12" skillet over medium-high heat, heat 2 Tbsp. oil. Working in batches, cook shrimp, making sure shrimp is touching skillet, until browned and cooked through, about 1 minute per side. Transfer to a plate. Wipe out skillet.
In a food processor, pulse onion, bell pepper, and garlic, scraping down sides if needed, until very smooth and a puree forms, about 1 minute.
In a large pot of boiling salted water, cook pasta, stirring occasionally, until al dente according to package directions. Reserve 1 cup pasta water, then drain.
Meanwhile, in same skillet over medium heat, heat remaining 2 Tbsp. oil. Cook onion mixture, stirring frequently and adding a splash of water if browning too quickly, until most of the liquid is evaporated and mixture becomes drier and thicker, 7 to 10 minutes.
Add cream to onion mixture, scraping any bits stuck to bottom of pan. Bring to a boil, then remove from heat. Whisk Parmesan into sauce until thick and homogenous. Add shrimp, pasta, parsley, chives, and ¾ cup pasta water, stirring and adding more pasta water if needed, until sauce is smooth and glossy. Top with more Parmesan.