Place a small saucepan over medium-high heat. Add the raspberries, sugar, and water to the pan, stirring to combine. Bring the mixture to a simmer, reduce heat, and allow to cook for five minutes. Remove the pan from the heat and add the mint sprigs to the pan. Steep the mint, stirring gently, about 30 seconds. Remove the mint. If desired, strain out the raspberry seeds with a fine mesh sieve. Allow the raspberry mint mixture to chill in the refrigerator.
Once cool, stir in the lime juice. Fill a small glass with ice cubes. Pour the raspberry-mint-lime mixture over ice, filling the cup about half to two-thirds full. Top off the glass with club soda, giving the drink a quick stir. Taste for sweetness, adding more raspberry or club soda as desired. Repeat steps with the remaining glasses.
Enjoy promptly with lime wedges, fresh raspberries, and/or mint on top. Recipe adapted from Delish Knowledge.