In medium bowl, whisk together flour, cocoa, baking soda, and salt.
Using electric mixer, beat butter and granulated sugar in large bowl until fluffy, about 3 minutes. Beat in eggs 1 at a time, then peppermint. Reduce speed to low and gradually add flour mixture, mixing until incorporated. Divide dough into 4 portions.
Roll 1 portion between 2 sheets parchment or waxed paper to 1/8 inch thick. Roll remaining portions to 1/4 inch thick. From 1/8-inch-thick dough, cut out doors, shutters, trailer fin, wheels, antler mount, and other smaller pieces. From 1/4-inch-thick dough, cut out remaining house pieces. Refrigerate until firm, at least 20 minutes, then pop out pieces. Heat oven to 350°F. Arrange like-size pieces on parchment-lined baking sheets, spacing 1 inch apart. Reroll, cut and chill as necessary.
In batches, bake cookies, rotating baking sheets halfway through, until firm, 8 to 15 minutes. (depending on thinness and size of cookies). Let cool on baking sheets 5 min., then transfer to wire rack to cool completely.
Reduce oven temp to 175°F. Arrange cookies on wire racks about 1/4 inch apart. Transfer to oven and allow to dry 1 hour.
Decorate pieces of house, then “glue” together with royal icing.