Heat oven to 350°F. Line two baking sheets with parchment.
Cream butter, granulated sugar, and brown sugar until pale and fluffy, 2–3 minutes.
Beat in egg and vanilla until smooth, scraping the bowl as needed.
Whisk flour, cocoa, baking soda, and salt. Add to the bowl and mix just combined. Splash in milk if dough looks dry.
Scoop dough into 1½-tablespoon mounds, spacing 2 inches apart. Slightly flatten the tops.
Bake 8–10 minutes until set at edges and soft in centers. Cool on sheet 5 minutes, then move to a rack to cool completely.
Make filling: Beat butter until creamy. Mix in marshmallow creme, powdered sugar, vanilla, and salt until fluffy, 1–2 minutes.
Spread or pipe a rounded teaspoon of filling onto each cooled cookie, leaving a small border. Chill 10 minutes to firm.
Make glaze: Pour warm cream over chocolate chips. Add corn syrup and salt. Let sit 2 minutes, then stir until glossy and smooth.
Spoon or dip glaze over the filled cookies to cover the tops. Let excess drip off. Set until the chocolate is tacky, 10–15 minutes.
Stir icing ingredients until thick but pipeable. Pipe a neat white squiggle across each cookie. Let set before serving.