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Cook pasta in boiling salted water according to package directions for al dente. Drain.
Meanwhile, toss together cheddar, Gruyère, and Parmigiano-Reggiano until well combined; set aside 1 cup cheese mixture. Preheat broiler to low with oven rack in top third of oven.
Bring cream just to a boil in a large saucepan over medium-high. Reduce heat to low; add garlic, and gently simmer, stirring often, 2 minutes. Stir in cooked pasta, and return to a gentle simmer over medium. Cook, stirring often, 1 minute. Reduce heat to medium-low. Stir in cheese mixture in three additions (about 1 ½ cups at a time), until cheese is melted after each addition. Gently stir in crab and ¼ cup chives. Season to taste with salt.
Spoon pasta mixture into an 8- x 6- x 2-inch broiler-proof baking dish set on a rimmed baking sheet. Top with reserved 1 cup cheese mixture in an even layer.
Broil in preheated oven until cheese is golden brown and crusty, 5 to 7 minutes, rotating as needed for even browning. Let stand 5 to 10 minutes, and sprinkle with remaining 1 tablespoon chives.