Slice 1 lime in half, then slice 2 wedges from 1 half and reserve for garnish. Juice remaining 1 1/2 limes to yield 2 Tbsp. juice. Combine lime juice, sugar, and 2 Tbsp. mint leaves in a measuring cup or cocktail shaker. Muddle mint with the back of a spoon. Let sit 5 minutes. Meanwhile, pour 1/2 cup water into a small bowl. Sift matcha through a fine-mesh sieve into water and whisk until smooth. Whisk matcha into mint mixture, then divide between 2 glasses filled with ice. Top with sparkling water. Garnish with lime wedges and remaining 1 Tbsp. mint.