Sauté shallot and garlic in oil in a large, heavy-bottomed pot over medium heat until very fragrant, 1 to 2 minutes. Add lemongrass and cook 1 to 2 more minutes. Pour in coconut milk, then stir in curry paste, fish sauce, and sugar. Increase heat to high and bring to a boil.
Add mussels, stir until well coated, and cover. Cook, stirring occasionally, until mussels open, 5 minutes. Stir in chopped basil and spoon into wide, shallow bowls, discarding any unopened mussels. Garnish with slivered basil and serve with lime.
*Find curry paste in your market's Asian-foods aisle. A mussel's "beard" is the clump of filaments it uses to attach itself to rocks.
Note: Nutritional analysis is per serving.