Christina's Pumpkin Whoopie Pies

1 1/2 sticks (6 ounces) unsalted butter, 1 stick melted, 1/2 stick softened
1 cup packed light brown sugar
2 large eggs, at room temperature, lightly beaten
1 cup canned pure pumpkin puree
1 tablespoon pumpkin pie spice
1 1/2 teaspoons pure vanilla extract, divided
1 teaspoon baking powder
1 teaspoon baking soda
1/2 teaspoon salt
1 2/3 cups flour
4 ounces cream cheese, chilled
1 cup confectioners' sugar

1. Preheat the oven to 350°. Line two baking sheets with parchment paper.
2. In a large bowl, whisk together the melted butter and brown sugar until smooth. Whisk in the eggs, pumpkin puree, pumpkin pie spice, 1 teaspoon vanilla, the baking powder, the baking soda and salt. Using a rubber spatula, fold in the flour.
3. Using an ice cream scoop, drop 12 generous mounds of batter, spaced evenly, onto each baking sheet. Bake until springy to the touch, about 10 minutes. Transfer to a rack to cool completely.
4. Meanwhile, using an electric mixer, cream the softened butter with the cream cheese. Add the confectioners' sugar and 1/2 teaspoon vanilla; mix on low speed until blended, then beat on medium-high speed until fluffy, about 2 minutes.
5. Spread the flat side of 12 cakes with the cream cheese frosting. Top each with another cake.