Add 1 cup granulated sugar and 1 cup water to a small saucepan and bring to a simmer over medium-high heat. When sugar is dissolved, add 2 cups sliced strawberries and 1 sprig fresh rosemary and return to simmer. Reduce heat to medium, and continue to simmer until strawberries have softened and begun to lose their color, about 10 minutes.
Allow to cool, then strain through a fine mesh strainer into a large measuring cup, gently pressing on the strawberries with a spoon to remove extra syrup. Pour into a glass bottle. Measure 1 Tablespoon balsamic vinegar into a 1 cup measuring cup. Add enough apple cider vinegar (about 15 Tablespoons) to equal 1 cup of vinegar total. Add the vinegar to the syrup in the bottle and shake gently to mix. Chill in the refrigerator for at least 2 hours before serving.
To serve: pour about 1/4 cup of the syrup over ice and dilute with cold still or sparkling water to taste. Shrub syrup can also be used to make cocktails and tastes especially good with vodka or gin, or mixed with white wine or sparkling wine.