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In a large pot of salted boiling water, cook spaghetti according to package directions until al dente. Drain, reserving 1 cup pasta water. Meanwhile, in a large skillet over medium heat, heat oil. Add garlic and cook until fragrant, 1 minute, then add heavy cream, broth, and 1/2 cup of reserved pasta water. Add cooked spaghetti and toss until noodles are fully coated and liquid is simmering. Remove from heat and stir in cheese. Toss constantly, adding more pasta water until sauce reaches desired consistency. Season with salt and pepper and garnish with chives before serving.