In a small saucepan on medium heat, combine sugar, water and rosemary sprigs. Stir until sugar dissolves. Bring to a simmer and remove from heat. Let cool completely. Store in a glass jar for up to 1 month.
In two copper mule mugs or highball glasses, add rosemary simple syrup, vodka and 3 kumquat halves. Muddle the kumquats, mashing and squishing them so the juices and skins infuse their flavor into the vodka and syrup.
Fill the glass completely with ice. Top with ginger beer. Garnish with a rosemary spring and a kumquat on the rim of the mug.