Heat the oil in a large saucepan over medium heat. When the oil is shimmering, add the garlic, ginger, shallot, and lemongrass and cook, stirring until the vegetables are softened but not browned, about 2 minutes.
Add the pepper and cook, stirring, until fragrant, about 2 minutes.
Pour in the coconut milk, hoisin, vinegar and fish sauce. Stir, bring to a boil and then lower the heat and simmer for 15 minutes to develop the flavor. Stir in the lime juice. The sauce can be refrigerated in an airtight container for up to 3 days.