Bring a large pot of salted water to a boil. Cook the pasta according to package instructions. In the last 3 minutes of cooking, add the cauliflower. Reserve 1 cup of the pasta cooking water. Drain the pasta and vegetables and return to the pot.
Meanwhile, in a small saucepan over medium, melt 1 tablespoon of butter. Add the shallot and cook, stirring frequently, until softened, about 2 minutes. Add the cream, reduce the heat to low, then cook without boiling, stirring occasionally, until warmed through and slightly thickened,about 2 minutes.
Stir in the Parmigiano-Reggiano and 1 tablespoon of butter. Taste and season with salt, pepper, nutmeg and more Parmigiano, as needed.
Scrape the sauce into the pot with the cooked pasta and cauliflower, splashing in some of the reserved pasta water as needed to thin out the sauce. Add the remaining 1 tablespoon of butter. Over medium heat, toss until the sauce thickens slightly and clings to the pasta. Taste the pasta once more and season with salt, if it needs it.
Serve immediately with a generous sprinkling of chives. Optionally, you can also enjoy this creamy pasta with dollops of cool ricotta, a squeeze of fresh lemon or a handful of arugula on top.