Add chopped peaches, cinnamon sticks, nutmeg, allspice, water, and brown sugar to a medium pot.
Place over medium heat and stir until the sugar dissolves.
Once dissolved, reduce to a gentle simmer and cook for 12–15 minutes.
Remove from heat, let cool, and discard cinnamon sticks.
Blend until smooth. Store in the fridge for up to 2 weeks.