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For the cookies:
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Preheat oven to 375 degrees.
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In a bowl, add flour, baking powder, baking soda, salt, and pumpkin pie spice. Whisk until blended.
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In another large bowl, mix butter and brown sugar with an electric mixer until light and fluffy. Mix in eggs, canned pumpkin, evaporated milk, and vanilla.
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Add the flour mixture to the wet pumpkin mixture. Mix until well combined.
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Spoon about a tablespoon of dough onto a parchment paper-lined baking sheet, spacing cookies 1 inch apart. Bake cookies for 12 minutes. Cool cookies completely before moving on to the sandwich making process.
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For the sandwich filling, mix butter with an electric mixer on low speed until fluffy. Stop the mixer and add the sugar, milk, and vanilla. Blend on low until the ingredients are incorporated. Increase the speed to medium and beat until light and fluffy. If frosting is too stiff or thick, add another tablespoon of milk.
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Take a cooled cookie and apply a teaspoon of frosting to the bottom of it. Then take another cookie and place it bottom down on top of the frosting. Gently squeeze the cookies together until the frosting has spread to the end of the cookie sides.
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Repeat this process until all your cookie sandwiches are made.
