Combine all ingredients in a shaker. Dry shake (no ice) for about 30-45 seconds, add ice, shake again until chilled. Strain into something pretty.
Combine all ingredients in a shaker with ice. Shake until slightly frothy and strain into a martini glass. Garnish with pineapple. (We also used some Coconut Lime finishing salt from Salinity Salts.)
For the charred pineapple purée:
Grill fresh pineapple slices (stove top grilling, barbecue grill or even a George Forman will work) until part of the pineapple is charred and the juices are bubbling. Let cool. Combine equal parts grilled pineapple and pineapple juice in a blender with juice of half a lime and blend on high until smooth.