Bring a large pot of salted water to a boil and cook the spaghetti 2 minutes shy of al dente according to the package. Reserve 3/4 cups of the hot pasta water and drain the noodles.
While the pasta is cooking, melt the butter in a large skillet or pan over medium heat. Add the garlic slices and cook, stirring, until the garlic is aromatic but not brown.
Carefully add a 1/2 cup of the pasta water to the skillet (it will bubble up, so be careful) and bring to a boil over high heat. Whisk the pasta water, butter, and garlic together, to emulsify into a loose sauce.
Add the drained spaghetti to the pan and continue to cook over medium-high heat, tossing occasionally, until the pasta is al dente and the liquid reduces to a glossy, thick sauce, about 2-3 minutes.
Stir in the cheese, tossing to melt and combine, thinning with extra pasta water if needed. Taste and season with salt and freshly ground pepper.
Divide the garlicky pasta between to plates and finish with a generous sprinkle of thinly sliced chives and dollops of caviar. Enjoy hot.