Preheat oven to 350° and line muffin tray with paper liners.
Place zest into bowl with brown sugar and use fingertips to rub zest into sugar. Once well combined, add the juice, coconut milk, eggs and Lime oil and whisk until smooth.
Pour batter into the prepared muffin tray and bake for 15 min until golden brown and tester comes out clean. Leave to cool in pan for 10 min on a wire rack before removing.
To make the frosting, beat the butter and lime zest in an electric mixer with paddle attachment at medium speed until creamy. Switch to low speed and gradually add confectioner's sugar alternately with the lime juice, syrup and coconut milk until blended smooth.