Bring a large pot of generously salted water to a boil. Cook the rigatoni until al dente according to the package directions. Reserve 1 cup of the pasta water before draining.
Meanwhile, season the shrimp with salt and pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the shrimp and cook for 1 to 2 minutes per side, or until just opaque and cooked through. Transfer to a plate.
Add the garlic to the same skillet and sauté for about 30 seconds, adding a little more olive oil if needed. Reduce the heat to medium, stir in the gochujang, and cook, stirring constantly, for about 1 minute.
Pour in the heavy cream and bring to a gentle simmer. Cook for 2 to 3 minutes, or until the sauce has thickened slightly.
Add the cooked pasta along with about ¼ cup of the reserved pasta water, tossing to coat. Add more pasta water if needed, a couple of tablespoons at a time, until the sauce reaches your desired consistency.
Stir in the butter until melted. Stir in the Parmesan.
Return the shrimp to the skillet and stir in the lemon juice. Taste and season with additional salt and pepper if needed.
Garnish with chopped chives and more Parmesan before serving.