Beat butter and brown sugar together in a bowl with a mixer on medium speed. Mix in molasses, then egg, until blended, scraping bowl as needed.
In a small bowl, combine flour, baking soda, salt, and spices; add to butter mixture on low speed, mixing until combined.
Divide dough in half, shape each into a disk, wrap in plastic wrap, and chill until firm, about 3 hours.
Preheat oven to 350°. Unwrap dough. On a generously floured surface, roll out each disk until 1/8 in. thick. Using round 2 1/2- to 3-in. cookie cutters, cut out dough and arrange circles about 1 in. apart on parchment-lined baking sheets (use a small spatula to transfer). Reroll scraps as needed.
Bake cookies until dry-looking and just starting to brown on edges, about 8 minutes. Set pans on racks and let cookies cool on pans.
Scoop about 2 tbsp. ice cream onto the flat side of one cookie and sandwich with another cookie. Freeze airtight until firm.
Make ahead: Chill dough (step 3) up to 1 week. Store baked cookies airtight up to 3 days.
Note: Nutritional analysis is per sandwich cookie.