Place a 9"x 5"x 3" metal loaf pan or other similarly sized container in the freezer. (1.5- to 2-quart capacity)
Whisk together the condensed milk, cocoa powder, and vanilla in a large bowl.
Place the cream in a separate bowl. Use an electric mixer with a whisk attachment to whip the cream at medium-high speed until soft peaks form.
Fold about 1 cup of the whipped cream mixture into the condensed milk mixture. Transfer that mixture to the whipped cream, and gently fold until combined. Stir in the brownie pieces. If you like, reserve a handful of brownie pieces for the top of the ice cream.
Transfer the mixture to the chilled loaf pan. If you reserved any brownie pieces, sprinkle them over the top of the ice cream.
Cover tightly and freeze for 5 hours, or until the ice cream has a scoopable texture.
Notes
Keep the ice cream well-sealed in the freezer. It should keep up to 2 weeks, but will be best within a week.