Melt butter in a 12-in. frying pan over medium heat. Add green garlic and cook, stirring often, until softened and fragrant, 4 to 5 minutes. Add wine, increase heat to medium-high, and cook until it has nearly evaporated, about 1 minute. Stir in cream, broth, parmesan, nutmeg, and 1 tsp. each salt and pepper. Bring to a simmer, reduce heat to medium-low, and cook until cream is slightly thickened, about 3 minutes.
Cook pasta according to package instructions until just tender, 3 to 4 minutes. Reserve 1 cup cooking water, then drain pasta.
Meanwhile, add shrimp to cream sauce, increase heat to medium, and cook, stirring occasionally, just until pink, 3 to 4 minutes. Add pasta to frying pan and stir to coat. Stir in reserved pasta water (mixture will be soupy, then thicken as it stands) and toss in chives. Add more salt and pepper if you like.
Tips for cooks
Green garlic is simply immature regular garlic and tastes fresher and more delicate. It has juicy, edible stalks with no bulbs at the base (if picked quite young), or bulbs with tiny cloves that don't need peeling.
FIND Look for slender pale green stalks in springtime at farmers' markets and some supermarkets. Or substitute green onions plus regular garlic (see the recipe).
PREP Trim green garlic as you would green onions, removing tough tops (save to flavor soup) and roots, plus any tough outer layers. Ten stalks yield 2 cups chopped.
*If you can't find green garlic, use 1 cup slivered green onions plus 3 minced garlic cloves.