Muddle the blackberries with the simple syrup in a pint glass. Add ice, bourbon and vermouth. Top off with club soda and stir to combine.
Combine 3/4 cup water with the sugar, peppercorns, thyme and lemon in a small saucepan over medium-high heat. When the sugar dissolves and mixture just begins to boil, lower the heat and simmer for 5 minutes. Remove from the heat and let cool, then strain through a fine-mesh sieve. Refrigerate in an airtight container.