Add the water, chopped peaches, brown sugar, cinnamon sticks, nutmeg, ginger, and allspice to a small saucepan.
Stir together and cook on medium heat, stirring until the sugar dissolves. Once it's dissolved, bring up to an ALMOST-boil, then turn down the heat and let it simmer for 12 to 15 minutes.
Remove from the heat and mash or muddle the peaches to squeeze out the juices and pulp, and then add the vanilla, almond extract, and a small squeeze of lemon juice. Stir and taste to make sure the balance is right.
Then remove the cinnamon sticks, and strain into a container. I used a fine mesh strainer. If you really care about getting all the particulates out, you could also pass through a coffee filter or paper towel, but I don't think this is necessary. Save the leftover peach mush, would be great on yogurt or oatmeal as a compote!
This should keep in the fridge for a week or two, but likely not much longer.