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Make the pumpkin spice syrup: Combine the water, brown sugar, pumpkin puree, and pumpkin pie spice in a saucepan.
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Bring the mixture to a boil over medium heat, stirring until the sugar dissolves.
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Reduce the heat and simmer for 5–10 minutes.
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Remove from heat and stir in the vanilla extract. Let the syrup cool, then strain it through a fine-mesh sieve to remove any solids. Store the syrup in a jar in the refrigerator.
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Make the latte: Fill a glass halfway with ice.
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Pour in the cooled coffee.
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Add 2–4 tablespoons pumpkin spice syrup, adjusting to taste.
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Pour in the milk and stir well.
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Top with whipped cream and a sprinkle of pumpkin pie spice, if desired.