Combine the plum, sugar, water, rosemary, orange slice, and cinnamon sticks in a small saucepan over medium heat. Stir until the sugar has dissolved, then bring to a gentle simmer.
Reduce the heat and simmer for about 8–10 minutes, gently pressing the plum pieces as they soften. Remove from the heat and let steep for 15–20 minutes.
Strain through a fine-mesh sieve, discarding the solids. Let cool completely before transferring to an airtight container. Store refrigerated for up to 2 weeks.
Cocktail
Add the chopped plum and sage leaves to a cocktail shaker and gently muddle to release the plum juices and sage oils.
Add the gin, almond liqueur, lemon juice, plum syrup, and egg white. Dry shake vigorously without ice for about 15 seconds to emulsify the egg white.
Add ice and shake again until well chilled. Double strain evenly between two chilled Nick & Nora or coupe glasses. Garnish each with a thin plum slice floated on top.