Lemon Thyme Simple Syrup:
1 cup water
1 cup white sugar
3 sprigs lemon thyme
1/2 fluid ounce lemon juice
5 fresh raspberries, divided
1 1/2 fluid ounces raspberry vodka
1 cup lime-flavored sparkling water (such as La Croix®), or as needed
1/2 slice lemon
1 sprig lemon thyme
Heat water, sugar, and 3 sprigs lemon thyme in a small saucepan, stirring, until bubbles form and sugar is dissolved, about 5 minutes. Remove from heat and let steep, about 10 minutes. Strain out lemon thyme.
Pour 1/2 fluid ounce of the lemon thyme syrup into a tall glass. Add lemon juice and 3 raspberries; mash with a muddler or wooden spoon. Fill glass with ice and vodka; stir vigorously with a long spoon until well chilled. Fill glass with sparkling water.
Garnish with remaining raspberries, lemon slice, and 1 lemon thyme sprig.