For the mint simple syrup: Combine the sugar and 1/2 cup water in a small saucepan over medium heat. When the sugar dissolves, turn off the heat, add the mint sprigs and let cool. Strain into a clean jar. Can be stored in the refrigerator for a week.
For the gin and tonic: Heat a grill pan and grill the lime slices on both sides, just until it gets nice grill marks but is still firm.
Cut 1 lime slice into pieces and put it into the bottom of a cocktail shaker or mixing glass along with the chopped cucumber, bitters and 1 tablespoon mint simple syrup. Muddle to mash all the solids. Add ice to the glass and pour over the gin. Shake or stir to get the gin cold. Strain into a martini glass (or over ice in a tall glass) and top with a splash of tonic water. Garnish with the grilled lime slice and cucumber spear.