Cube your watermelon into 1-inch chunks, removing any seeds. Place the watermelon in a blender and pulse until you achieve a smooth consistency.
In two tall glasses, add 5-6 mint leaves and 2 lime quarters each. Using a muddler (or the back of a wooden spoon), gently press the mint and lime to release their essential oils and juices.
Add 1 tablespoon of sugar or simple syrup to each glass. If you're using honey, warm it slightly first to help it incorporate better.
Pour the watermelon puree evenly between the two glasses (about ½ cup per glass). Stir gently to combine with the mint, lime, and sweetener.
For alcoholic versions, add 1 oz of white rum to each glass. Fill the glasses about ¾ full with ice cubes.
Top each glass with club soda or sparkling water. Gently stir to combine all ingredients without crushing the ice. Garnish with a sprig of fresh mint and a small watermelon wedge on the rim of the glass.