Preheat a pizza oven or regular oven to 500 degrees F.
For the shrimp: Clean the shrimp, reserving the shells, and cut in half lengthwise. Evenly sprinkle the Cajun seasoning and butter slabs in the bottom of a medium cast-iron pan. Add the prepped shrimp and transfer to the oven to cook, stirring halfway through, about 4 minutes.
For the sauce: Add the oil, reserved shrimp shells, onions, celery, carrot, serrano pepper, tomato paste and basil to a medium saucepan over medium heat. Once the tomato paste is incorporated, add the wine and reduce by half. Strain into another saucepan over medium-low heat and stir in the butter. Taste and adjust the seasoning.
For the carrot butter: Add the carrots, butter and salt to a saucepan and allow to slowly simmer until the carrots are soft, about 15 minutes. Pour into a blender and blend until smooth. Set aside.
For the chimichurri: Add the carrot tops, parsley, brunoise carrot, vinegar and oil to a small bowl. Season with salt and stir to fully combine. Refrigerate 30 minutes.
Assemble: Place the carrot butter purée at the bottom of a plate, then place the shrimp sauce around the carrot purée. Top the shrimp with the chimichurri.
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