Combine 1 cup water and 1 cup sugar in a small saucepan and bring to a simmer, stirring until the sugar dissolves.
Add 1 tablespoon dried lavender buds to the pan, remove from heat, and let steep for 20 minutes.
Strain the syrup through a fine mesh strainer or cheesecloth to remove the lavender buds. Let the lavender syrup cool completely.
In a large pitcher, mix the fresh lemon juice, 2 cups cold water, and all of the cooled lavender syrup.
Fill glasses with ice cubes. Pour the lavender lemonade mixture into each glass, filling about three-quarters full.
Top off each glass with chilled club soda or sparkling water.
Stir gently, then garnish with lemon slices and, if desired, a sprig of lavender.