Place the coconut milk can or carton in the refrigerator to chill it.
In a high-speed blender, purée 1-1/4 cups of the fresh strawberries.
In a saucepan over medium-high heat, combine the puréed strawberries and sugar. Stir until it comes to a boil. Cook for about 1 minute until the sugar has dissolved and the purée has thickened slightly. Remove the pot from the heat and allow the purée to cool. Or, chill it in the refrigerator to speed up the process.
Grab two tall cocktail glasses. Spoon 1/2 of the cooled purée into each glass. Then, add 1/2 of the sliced strawberries to each glass. Rotate and tilt the glasses to coat the sides with the purée and strawberries.
Pour 1 cup of cold coconut milk into each glass. Swirl gently with a spoon just to get a little pattern of red streaking through the milk. Don’t stir it completely. This should look like a free-form painting, not a pink milkshake.
Garnish each glass with a mint leaf and serve immediately with tall spoons.