In a Vitamix or other high-speed blender, combine rum, lime juice and Thai basil. Blend on low speed for 1 to 2 seconds, just long enough to break up the leaves. Switch blender to highest setting and blend for 3 additional seconds. Strain through a fine mesh strainer into a chilled highball or pilsner glass filled halfway with crushed ice. Add simple syrup, salt and bitters.
In glass, twist a spoon or swizzle stick until cocktail is mixed. Top with more crushed ice and twist spoon until the glass frosts over. Garnish with remaining whole Thai basil and serve with a straw.