Preheat oven to 350 degrees, with racks in upper and lower thirds. In a medium bowl, whisk together flours, salt, baking soda, baking powder, and cinnamon. In a large bowl, using an electric mixer, beat butter and brown sugar on high, scraping down bowl as needed, until light, 4 minutes. Add egg and beat to combine, scraping down bowl as needed. With mixer on low, gradually add flour mixture and beat until combined.
Drop dough by tablespoonfuls, 3 inches apart, onto two parchment-lined baking sheets. With the bottom of a measuring cup dipped in flour, flatten cookies to 2 inches in diameter. Bake until dry and set, about 7 minutes, rotating sheets halfway through. Remove from oven and top each cookie with a marshmallow half. Bake until marshmallows are soft, 2 minutes.
Remove from oven and, with a metal spatula, gently flatten each marshmallow. Let cookies cool completely on sheets on wire racks.
Place chocolate in a medium heatproof bowl set over (not in) a saucepan of simmering water. Stir until melted, 2 minutes. Remove bowl from pan. Place one cookie at a time on tines of a fork, submerge in chocolate, then tap fork on edge of bowl to remove excess. Place on wire rack set over a baking sheet. Let cookies set in refrigerator, about 10 minutes.
Microwave chocolate in 30-second increments, stirring after each, until melted and smooth.