In the bowl of a blender or food processor, puree the mascarpone, milk, cream, sugar, salt, and vanilla until smooth, scraping down the sides of the bowl as necessary.

Transfer the mixture to a medium bowl and press plastic wrap against the surface.

Refrigerate until completely chilled, at least 30 minutes.

Pour the chilled mixture into an ice cream maker. Freeze according to the manufacturer’s directions. During the last 5 minutes of freezing add the chocolate chunks and rum. Transfer the ice cream to an airtight container, press plastic wrap against the ice cream surface, and freeze until it is firm and the flavor is ripened, at least 4 hours.

Preheat the oven to 350°F.

Line baking sheets with parchment paper or silicone baking mats.

In a medium bowl, whisk together the flour, espresso powder, baking soda, and salt.

In a large bowl, use an electric mixer to beat the butter and sugars on medium-high speed until well combined and smooth, about 1 to 2 minutes. Add the milk, egg, egg yolk, and vanilla and beat until combined. On low speed, gradually add the flour mixture and beat until just combined.

Using a spoon or spring-loaded scoop, drop 2 tablespoon–sized balls of dough onto pre- pared baking sheets.

Bake for about 10 minutes, or until the edges are slightly browned. Let cool on baking sheets for 5 minutes before removing to wire racks to cool completely. Freeze the cookies until frozen, at least 1 hour. Cookies can be stored in airtight containers in the freezer for up to 1 month.

To assemble, top one cookie with a scoop of ice cream. Place another cookie on top of the ice cream and gently press down to form a sandwich. Immediately place the sandwich in the freezer. Repeat for the remaining cookies. Freeze for at least 2 hours.