Preheat the oven’s broiler and set a rack about 6 inches below the heat source. Line a baking sheet with parchment, and place marshmallows on the baking sheet.
Broil until marshmallow tops are well browned, 1 to 2 minutes.
Add water and sugar to a saucepan over medium-high heat. Bring to a simmer and cook, stirring often, until sugar dissolves.
Reduce heat; stir in marshmallows, salt, and vanilla and cook, stirring constantly, until creamy.
Strain mixture through a fine sieve and stir in heavy cream. Serve immediately, stirred into coffee.
