Bring a large pot of salted water to a boil over medium-high. Add lobster tails; cook, undisturbed, until shells turn red but meat is still slightly translucent, about 2 minutes (lobster should be undercooked). Using tongs, transfer lobster to a large bowl; let cool 5 minutes. Using kitchen shears and working over the bowl to catch any juices, cut down the center of each lobster tail shell; transfer meat to a cutting board. Discard shells and juices. Remove and discard digestive tract from lobster tails. Cut lobster meat crosswise into ½-inch-thick slices, and place in a medium bowl; cover and chill until ready to use.

Return lobster cooking water to a boil over high. Add pasta, and cook until al dente, about 12 minutes; drain, reserving ¾ cup cooking liquid. Set aside pasta and reserved cooking liquid.

Melt butter in a large skillet over medium. Add garlic, crushed red pepper, and 2 cups of the tomatoes; cook, stirring occasionally, until tomatoes are softened and burst, about 5 minutes.

Add cream, oregano, salt, and black pepper to skillet; bring to a simmer over medium. Gradually add spinach; cook, stirring often, until spinach is wilted, 1 to 2 minutes.

Increase heat in skillet to medium-high. Stir in cooked pasta and remaining 2 cups tomatoes. Cook, stirring constantly, until sauce thickens enough to coat pasta and tomatoes begin to soften, about 3 minutes. Gradually stir in reserved cooking liquid, as needed, to reach desired consistency. Remove from heat, and stir in lobster and Parmesan until well combined and cheese is melted, about 1 minute.

Divide pasta evenly among 6 bowls; sprinkle with additional Parmesan, and garnish with chives.