Bring a large pot of water to a boil. Season with salt until it tastes like the sea. Cook the pasta until not quite al dente, about 7 minutes. Transfer with a spaghetti scoop or slotted spoon to a bowl and set aside. Add the broccoli to the pasta water and blanch briefly. Transfer with a slotted spoon to a bowl. Reserve the pasta water.
While the pasta is cooking, heat the butter in a large pan over medium heat. Once the butter is melted, add the garlic and cook until fragrant, about 30 seconds. Whisk in the flour and cook for 2 minutes. Whisk in the milk, then bring to a boil. Reduce the heat to a simmer and cook, whisking, until the sauce is slightly thickened, about 5 minutes. Stir in the Gruyere, sour cream and Parmesan and whisk until smooth. Add the hot sauce and season with salt and pepper.
Add the pasta, chicken and broccoli to the sauce. Continue to cook until the pasta is al dente and the sauce coats the pasta, 4 to 5 minutes. Use pasta water to adjust the consistency of the sauce if needed.
Distribute the pasta into bowls and top with the bacon and chives.