Place butter into a light-colored saucepan over medium heat. Cook, stirring frequently, until butter is browned and smells nutty, about 5 to 10 minutes. Pour butter into a large mixing bowl.

In the same saucepan, carefully pour in 2 cups of apple cider and cook until reduced to 1/2 cup, about 20 to 25 minutes. Reserve 2 tablespoons for the icing and pour the remaining reduced cider into the bowl with browned butter. Allow mixture to cool for 20 minutes.

Add sugars, egg, vanilla, cinnamon, salt, nutmeg, cloves, and allspice to the butter-cider mixture; mix thoroughly. Incorporate half the flour, cornstarch, and baking powder. Mix in the remaining flour until just combined. Knead dough into a disc, wrap, and refrigerate for at least 1 hour (up to 48 hours).

On a floured surface, roll out dough to 1/4-inch thick, cut out cookies using cutters, and gather scraps to roll and cut again. Leave cut-outs to chill for at least 15 minutes in the refrigerator.

Preheat the oven to 350°F (175°C). Transfer chilled cookie cut-outs to lined baking sheets, spacing them 1 inch apart.

Bake cookies for 10 to 12 minutes or until slightly puffed. Let cool on the sheet for several minutes before transferring to a wire rack until fully cooled.

Prepare icing by whisking reserved reduced cider, remaining apple cider, heavy cream, confectioner's sugar, honey, vanilla extract, and salt until smooth. Adjust consistency as needed.