Put mint leaves in a large heatproof bowl. Pour boiling water over mint and let steep 10 minutes. Strain into a pitcher or another bowl and let the mint tea cool to room temperature.
Meanwhile, in a small saucepan over medium-high heat, bring cranberries, sugar, and 1 cup water to a boil. Remove from heat and let cool to room temperature. Strain the cranberry syrup into another pitcher or bowl.
For each cocktail, combine 1/3 cup mint tea, 2 tbsp. cranberry syrup, 1 tbsp. limoncello, 2 tbsp. vodka, and 1/2 tsp. lemon juice in a cocktail shaker full of ice. Shake vigorously and strain into a martini glass. Garnish with a sprig of mint.