Preheat oven to 400 degrees. Line a sheet pan with foil. Cut peaches in half and place on pan, cut side up. Roast in oven until fruit is very soft and slightly charred, about 25 minutes. Remove from oven and let cool slightly.
Once fruit is cool enough to handle, spoon out pits and slip skins off.
In a heavy-bottomed saucepan over medium heat, soften onion and garlic in oil. Add peaches, cinnamon stick, paprika, brown sugar, cayenne, water, and worcestershire. Simmer until all flavors meld, about 20 minutes, stirring often to prevent scorching. Remove from heat and let cool about 10 to 15 minutes.
Remove cinnamon stick, place mixture in blender, and purée until very smooth. Lightly season with salt and pepper and chill for several hours. Keeps in refrigerator for 2 weeks.