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Preheat a waffle iron according to manufacturer's instructions.
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Combine flour, brown sugar, baking powder, cinnamon, allspice, ginger, and salt in a mixing bowl. Stir milk, pumpkin, and egg yolks together in a separate bowl. Whip egg whites in a clean dry bowl until soft peaks form.
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Stir flour mixture and 1/4 cup melted butter into pumpkin mixture just until combined. Fold 1/3 of the egg whites into the batter with a spatula until incorporated. Gently fold in remaining egg whites. Cook waffles in preheated waffle iron according to manufacturer's instructions.
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To make the syrup: Stir sugar, cornstarch, and cinnamon together in a saucepan. Stir in apple cider and lemon juice. Cook over medium heat until mixture comes to a boil; continue boiling until thickened, about 2 minutes. Remove from heat and stir in 2 tablespoons of butter until melted. Serve waffles warm with apple cider syrup.